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{{task}} A given rectangle is made from ''m'' × ''n'' squares. If ''m'' and ''n'' are not both odd, then it is possible to cut a path through the rectangle along the square edges such that the rectangle splits into two connected pieces with the same shape (after rotating one of the pieces by 180°). All such paths for 2 × 2 and 4 × 3 rectangles are shown below.
[[file:rect-cut.svg]]
Write a program that calculates the number of different ways to cut an ''m'' × ''n'' rectangle. Optionally, show each of the cuts.
Possibly related task: [[Maze generation]] for depth-first search.
C
Exhaustive search on the cutting path. Symmetric configurations are only calculated once, which helps with larger sized grids.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
typedef unsigned char byte;
byte *grid = 0;
int w, h, len;
unsigned long long cnt;
static int next[4], dir[4][2] = {{0, -1}, {-1, 0}, {0, 1}, {1, 0}};
void walk(int y, int x)
{
int i, t;
if (!y || y == h || !x || x == w) {
cnt += 2;
return;
}
t = y * (w + 1) + x;
grid[t]++, grid[len - t]++;
for (i = 0; i < 4; i++)
if (!grid[t + next[i]])
walk(y + dir[i][0], x + dir[i][1]);
grid[t]--, grid[len - t]--;
}
unsigned long long solve(int hh, int ww, int recur)
{
int t, cx, cy, x;
h = hh, w = ww;
if (h & 1) t = w, w = h, h = t;
if (h & 1) return 0;
if (w == 1) return 1;
if (w == 2) return h;
if (h == 2) return w;
cy = h / 2, cx = w / 2;
len = (h + 1) * (w + 1);
grid = realloc(grid, len);
memset(grid, 0, len--);
next[0] = -1;
next[1] = -w - 1;
next[2] = 1;
next[3] = w + 1;
if (recur) cnt = 0;
for (x = cx + 1; x < w; x++) {
t = cy * (w + 1) + x;
grid[t] = 1;
grid[len - t] = 1;
walk(cy - 1, x);
}
cnt++;
if (h == w)
cnt *= 2;
else if (!(w & 1) && recur)
solve(w, h, 0);
return cnt;
}
int main()
{
int y, x;
for (y = 1; y <= 10; y++)
for (x = 1; x <= y; x++)
if (!(x & 1) || !(y & 1))
printf("%d x %d: %llu\n", y, x, solve(y, x, 1));
return 0;
}
output
More awkward solution: after compiling, run <code>./a.out -v [width] [height]</code> for display of cuts.
```c
#include <stdio.h>
#include <stdlib.h>
typedef unsigned char byte;
int w = 0, h = 0, verbose = 0;
unsigned long count = 0;
byte **hor, **ver, **vis;
byte **c = 0;
enum { U = 1, D = 2, L = 4, R = 8 };
byte ** alloc2(int w, int h)
{
int i;
byte **x = calloc(1, sizeof(byte*) * h + h * w);
x[0] = (byte *)&x[h];
for (i = 1; i < h; i++)
x[i] = x[i - 1] + w;
return x;
}
void show()
{
int i, j, v, last_v;
printf("%ld\n", count);
#if 0
for (i = 0; i <= h; i++) {
for (j = 0; j <= w; j++)
printf("%d ", hor[i][j]);
puts("");
}
puts("");
for (i = 0; i <= h; i++) {
for (j = 0; j <= w; j++)
printf("%d ", ver[i][j]);
puts("");
}
puts("");
#endif
for (i = 0; i < h; i++) {
if (!i) v = last_v = 0;
else last_v = v = hor[i][0] ? !last_v : last_v;
for (j = 0; j < w; v = ver[i][++j] ? !v : v)
printf(v ? "\033[31m[]" : "\033[33m{}");
puts("\033[m");
}
putchar('\n');
}
void walk(int y, int x)
{
if (x < 0 || y < 0 || x > w || y > h) return;
if (!x || !y || x == w || y == h) {
++count;
if (verbose) show();
return;
}
if (vis[y][x]) return;
vis[y][x]++; vis[h - y][w - x]++;
if (x && !hor[y][x - 1]) {
hor[y][x - 1] = hor[h - y][w - x] = 1;
walk(y, x - 1);
hor[y][x - 1] = hor[h - y][w - x] = 0;
}
if (x < w && !hor[y][x]) {
hor[y][x] = hor[h - y][w - x - 1] = 1;
walk(y, x + 1);
hor[y][x] = hor[h - y][w - x - 1] = 0;
}
if (y && !ver[y - 1][x]) {
ver[y - 1][x] = ver[h - y][w - x] = 1;
walk(y - 1, x);
ver[y - 1][x] = ver[h - y][w - x] = 0;
}
if (y < h && !ver[y][x]) {
ver[y][x] = ver[h - y - 1][w - x] = 1;
walk(y + 1, x);
ver[y][x] = ver[h - y - 1][w - x] = 0;
}
vis[y][x]--; vis[h - y][w - x]--;
}
void cut(void)
{
if (1 & (h * w)) return;
hor = alloc2(w + 1, h + 1);
ver = alloc2(w + 1, h + 1);
vis = alloc2(w + 1, h + 1);
if (h & 1) {
ver[h/2][w/2] = 1;
walk(h / 2, w / 2);
} else if (w & 1) {
hor[h/2][w/2] = 1;
walk(h / 2, w / 2);
} else {
vis[h/2][w/2] = 1;
hor[h/2][w/2-1] = hor[h/2][w/2] = 1;
walk(h / 2, w / 2 - 1);
hor[h/2][w/2-1] = hor[h/2][w/2] = 0;
ver[h/2 - 1][w/2] = ver[h/2][w/2] = 1;
walk(h / 2 - 1, w/2);
}
}
void cwalk(int y, int x, int d)
{
if (!y || y == h || !x || x == w) {
++count;
return;
}
vis[y][x] = vis[h-y][w-x] = 1;
if (x && !vis[y][x-1])
cwalk(y, x - 1, d|1);
if ((d&1) && x < w && !vis[y][x+1])
cwalk(y, x + 1, d|1);
if (y && !vis[y-1][x])
cwalk(y - 1, x, d|2);
if ((d&2) && y < h && !vis[y + 1][x])
cwalk(y + 1, x, d|2);
vis[y][x] = vis[h-y][w-x] = 0;
}
void count_only(void)
{
int t;
long res;
if (h * w & 1) return;
if (h & 1) t = h, h = w, w = t;
vis = alloc2(w + 1, h + 1);
vis[h/2][w/2] = 1;
if (w & 1) vis[h/2][w/2 + 1] = 1;
if (w > 1) {
cwalk(h/2, w/2 - 1, 1);
res = 2 * count - 1;
count = 0;
if (w != h)
cwalk(h/2+1, w/2, (w & 1) ? 3 : 2);
res += 2 * count - !(w & 1);
} else {
res = 1;
}
if (w == h) res = 2 * res + 2;
count = res;
}
int main(int c, char **v)
{
int i;
for (i = 1; i < c; i++) {
if (v[i][0] == '-' && v[i][1] == 'v' && !v[i][2]) {
verbose = 1;
} else if (!w) {
w = atoi(v[i]);
if (w <= 0) goto bail;
} else if (!h) {
h = atoi(v[i]);
if (h <= 0) goto bail;
} else
goto bail;
}
if (!w) goto bail;
if (!h) h = w;
if (verbose) cut();
else count_only();
printf("Total: %ld\n", count);
return 0;
bail: fprintf(stderr, "bad args\n");
return 1;
}
Common Lisp
Count only.
(defun cut-it (w h &optional (recur t))
(if (oddp (* w h)) (return-from cut-it 0))
(if (oddp h) (rotatef w h))
(if (= w 1) (return-from cut-it 1))
(let ((cnt 0)
(m (make-array (list (1+ h) (1+ w))
:element-type 'bit
:initial-element 0))
(cy (truncate h 2))
(cx (truncate w 2)))
(setf (aref m cy cx) 1)
(if (oddp w) (setf (aref m cy (1+ cx)) 1))
(labels
((walk (y x turned)
(when (or (= y 0) (= y h) (= x 0) (= x w))
(incf cnt (if turned 2 1))
(return-from walk))
(setf (aref m y x) 1)
(setf (aref m (- h y) (- w x)) 1)
(loop for i from 0
for (dy dx) in '((0 -1) (-1 0) (0 1) (1 0))
while (or turned (< i 2)) do
(let ((y2 (+ y dy))
(x2 (+ x dx)))
(when (zerop (aref m y2 x2))
(walk y2 x2 (or turned (> i 0))))))
(setf (aref m (- h y) (- w x)) 0)
(setf (aref m y x) 0)))
(walk cy (1- cx) nil)
(cond ((= h w) (incf cnt cnt))
((oddp w) (walk (1- cy) cx t))
(recur (incf cnt (cut-it h w nil))))
cnt)))
(loop for w from 1 to 9 do
(loop for h from 1 to w do
(if (evenp (* w h))
(format t "~d x ~d: ~d~%" w h (cut-it w h)))))
output
## D
{{trans|C}}
```d
import core.stdc.stdio, core.stdc.stdlib, core.stdc.string, std.typecons;
enum int[2][4] dir = [[0, -1], [-1, 0], [0, 1], [1, 0]];
__gshared ubyte[] grid;
__gshared uint w, h, len;
__gshared ulong cnt;
__gshared uint[4] next;
void walk(in uint y, in uint x) nothrow @nogc {
if (!y || y == h || !x || x == w) {
cnt += 2;
return;
}
immutable t = y * (w + 1) + x;
grid[t]++;
grid[len - t]++;
foreach (immutable i; staticIota!(0, 4))
if (!grid[t + next[i]])
walk(y + dir[i][0], x + dir[i][1]);
grid[t]--;
grid[len - t]--;
}
ulong solve(in uint hh, in uint ww, in bool recur) nothrow @nogc {
h = (hh & 1) ? ww : hh;
w = (hh & 1) ? hh : ww;
if (h & 1) return 0;
if (w == 1) return 1;
if (w == 2) return h;
if (h == 2) return w;
immutable cy = h / 2;
immutable cx = w / 2;
len = (h + 1) * (w + 1);
{
// grid.length = len; // Slower.
alias T = typeof(grid[0]);
auto ptr = cast(T*)alloca(len * T.sizeof);
if (ptr == null)
exit(1);
grid = ptr[0 .. len];
}
grid[] = 0;
len--;
next = [-1, -w - 1, 1, w + 1];
if (recur)
cnt = 0;
foreach (immutable x; cx + 1 .. w) {
immutable t = cy * (w + 1) + x;
grid[t] = 1;
grid[len - t] = 1;
walk(cy - 1, x);
}
cnt++;
if (h == w)
cnt *= 2;
else if (!(w & 1) && recur)
solve(w, h, 0);
return cnt;
}
void main() {
foreach (immutable uint y; 1 .. 11)
foreach (immutable uint x; 1 .. y + 1)
if (!(x & 1) || !(y & 1))
printf("%d x %d: %llu\n", y, x, solve(y, x, true));
}
{{out}}
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
10 x 1: 1
10 x 2: 10
10 x 3: 115
10 x 4: 1228
10 x 5: 10295
10 x 6: 118276
10 x 7: 1026005
10 x 8: 11736888
10 x 9: 99953769
10 x 10: 1124140214
Using the LDC2 compiler the runtime is about 15.98 seconds (the first C entry runs in about 16.75 seconds with GCC).
Eiffel
class
APPLICATION
create
make
feature {NONE} -- Initialization
make
-- Finds solution for cut a rectangle up to 10 x 10.
local
i, j, n: Integer
r: GRID
do
n := 10
from
i := 1
until
i > n
loop
from
j := 1
until
j > i
loop
if i.bit_and (1) /= 1 or j.bit_and (1) /= 1 then
create r.make (i, j)
r.print_solution
end
j := j + 1
end
i := i + 1
end
end
end
class
GRID
create
make
feature {NONE}
n: INTEGER
m: INTEGER
feature
print_solution
-- Prints solution to cut a rectangle.
do
calculate_possibilities
io.put_string ("Rectangle " + n.out + " x " + m.out + ": " + count.out + " possibilities%N")
end
count: INTEGER
-- Number of solutions
make (a_n: INTEGER; a_m: INTEGER)
-- Initialize Problem with 'a_n' and 'a_m'.
require
a_n > 0
a_m > 0
do
n := a_n
m := a_m
count := 0
end
calculate_possibilities
-- Select all possible starting points.
local
i: INTEGER
do
if (n = 1 or m = 1) then
count := 1
end
from
i := 0
until
i > n or (n = 1 or m = 1)
loop
solve (create {POINT}.make_with_values (i, 0), create {POINT}.make_with_values (n - i, m), create {LINKED_LIST [POINT]}.make, create {LINKED_LIST [POINT]}.make)
i := i + 1
variant
n - i + 1
end
from
i := 0
until
i > m or (n = 1 or m = 1)
loop
solve (create {POINT}.make_with_values (n, i), create {POINT}.make_with_values (0, m - i), create {LINKED_LIST [POINT]}.make, create {LINKED_LIST [POINT]}.make)
i := i + 1
variant
m - i + 1
end
end
feature {NONE}
solve (p, q: POINT; visited_p, visited_q: LINKED_LIST [POINT])
-- Recursive solution of cut a rectangle.
local
possible_next: LINKED_LIST [POINT]
next: LINKED_LIST [POINT]
opposite: POINT
do
if p.negative or q.negative then
elseif p.same (q) then
add_solution
else
possible_next := get_possible_next (p)
create next.make
across
possible_next as x
loop
if x.item.x >= n or x.item.y >= m then
-- Next point cannot be on the border. Do nothing.
elseif x.item.same (q) then
add_solution
elseif not contains (x.item, visited_p) and not contains (x.item, visited_q) then
next.extend (x.item)
end
end
across
next as x
loop
-- Move in one direction
-- Calculate the opposite end of the cut by moving into the opposite direction (compared to p -> x)
create opposite.make_with_values (q.x - (x.item.x - p.x), q.y - (x.item.y - p.y))
visited_p.extend (p)
visited_q.extend (q)
solve (x.item, opposite, visited_p, visited_q)
-- Remove last point again
visited_p.finish
visited_p.remove
visited_q.finish
visited_q.remove
end
end
end
get_possible_next (p: POINT): LINKED_LIST [POINT]
-- Four possible next points.
local
q: POINT
do
create Result.make
--up
create q.make_with_values (p.x + 1, p.y)
if q.valid and q.x <= n and q.y <= m then
Result.extend (q);
end
--down
create q.make_with_values (p.x - 1, p.y)
if q.valid and q.x <= n and q.y <= m then
Result.extend (q)
end
--left
create q.make_with_values (p.x, p.y - 1)
if q.valid and q.x <= n and q.y <= m then
Result.extend (q)
end
--right
create q.make_with_values (p.x, p.y + 1)
if q.valid and q.x <= n and q.y <= m then
Result.extend (q)
end
end
add_solution
-- Increment count.
do
count := count + 1
end
contains (p: POINT; set: LINKED_LIST [POINT]): BOOLEAN
-- Does set contain 'p'?
do
set.compare_objects
Result := set.has (p)
end
end
class
POINT
create
make, make_with_values
feature
make_with_values (a_x: INTEGER; a_y: INTEGER)
-- Initialize x and y with 'a_x' and 'a_y'.
do
x := a_x
y := a_y
end
make
-- Initialize x and y with 0.
do
x := 0
y := 0
end
x: INTEGER
y: INTEGER
negative: BOOLEAN
-- Are x or y negative?
do
Result := x < 0 or y < 0
end
same (other: POINT): BOOLEAN
-- Does x and y equal 'other's x and y?
do
Result := (x = other.x) and (y = other.y)
end
valid: BOOLEAN
-- Are x and y valid points?
do
Result := (x > 0) and (y > 0)
end
end
{{out}}
Rectangle 2 x 1: 1 possibilities
Rectangle 2 x 2: 2 possibilities
Rectangle 3 x 2: 3 possibilities
Rectangle 4 x 1: 1 possibilities
Rectangle 4 x 2: 4 possibilities
Rectangle 4 x 3: 9 possibilities
Rectangle 4 x 4: 22 possibilities
Rectangle 5 x 2: 5 possibilities
Rectangle 5 x 4: 39 possibilities
Rectangle 6 x 1: 1 possibilities
Rectangle 6 x 2: 6 possibilities
Rectangle 6 x 3: 23 possibilities
Rectangle 6 x 4: 90 possibilities
Rectangle 6 x 5: 263 possibilities
Rectangle 6 x 6: 1018 possibilities
Rectangle 7 x 2: 7 possibilities
Rectangle 7 x 4: 151 possibilities
Rectangle 7 x 6: 2947 possibilities
Rectangle 8 x 1: 1 possibilities
Rectangle 8 x 2: 8 possibilities
Rectangle 8 x 3: 53 possibilities
Rectangle 8 x 4: 340 possibilities
Rectangle 8 x 5: 1675 possibilities
Rectangle 8 x 6: 11174 possibilities
Rectangle 8 x 7: 55939 possibilities
Rectangle 8 x 8: 369050 possibilities
Rectangle 9 x 2: 9 possibilities
Rectangle 9 x 4: 553 possibilities
Rectangle 9 x 6: 31721 possibilities
Rectangle 9 x 8: 1812667 possibilities
Rectangle 10 x 1: 1 possibilities
Rectangle 10 x 2: 10 possibilities
Rectangle 10 x 3: 115 possibilities
Rectangle 10 x 4: 1228 possibilities
Rectangle 10 x 5: 10295 possibilities
Rectangle 10 x 6: 118276 possibilities
Rectangle 10 x 7: 1026005 possibilities
Rectangle 10 x 8: 11736888 possibilities
Rectangle 10 x 9: 99953769 possibilities
Rectangle 10 x 10: 1124140214 possibilities
Elixir
{{trans|Ruby}}
Count only
import Integer
defmodule Rectangle do
def cut_it(h, w) when is_odd(h) and is_odd(w), do: 0
def cut_it(h, w) when is_odd(h), do: cut_it(w, h)
def cut_it(_, 1), do: 1
def cut_it(h, 2), do: h
def cut_it(2, w), do: w
def cut_it(h, w) do
grid = List.duplicate(false, (h + 1) * (w + 1))
t = div(h, 2) * (w + 1) + div(w, 2)
if is_odd(w) do
grid = grid |> List.replace_at(t, true) |> List.replace_at(t+1, true)
walk(h, w, div(h, 2), div(w, 2) - 1, grid) + walk(h, w, div(h, 2) - 1, div(w, 2), grid) * 2
else
grid = grid |> List.replace_at(t, true)
count = walk(h, w, div(h, 2), div(w, 2) - 1, grid)
if h == w, do: count * 2,
else: count + walk(h, w, div(h, 2) - 1, div(w, 2), grid)
end
end
defp walk(h, w, y, x, grid, count\\0)
defp walk(h, w, y, x,_grid, count) when y in [0,h] or x in [0,w], do: count+1
defp walk(h, w, y, x, grid, count) do
blen = (h + 1) * (w + 1) - 1
t = y * (w + 1) + x
grid = grid |> List.replace_at(t, true) |> List.replace_at(blen-t, true)
Enum.reduce(next(w), count, fn {nt, dy, dx}, cnt ->
if Enum.at(grid, t+nt), do: cnt, else: cnt + walk(h, w, y+dy, x+dx, grid)
end)
end
defp next(w), do: [{w+1, 1, 0}, {-w-1, -1, 0}, {-1, 0, -1}, {1, 0, 1}] # {next,dy,dx}
end
Enum.each(1..9, fn w ->
Enum.each(1..w, fn h ->
if is_even(w * h), do: IO.puts "#{w} x #{h}: #{Rectangle.cut_it(w, h)}"
end)
end)
{{out}}
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
Show each of the cuts
{{works with|Elixir|1.2}}
defmodule Rectangle do
def cut(h, w, disp\\true) when rem(h,2)==0 or rem(w,2)==0 do
limit = div(h * w, 2)
start_link
grid = make_grid(h, w)
walk(h, w, grid, 0, 0, limit, %{}, [])
if disp, do: display(h, w)
result = Agent.get(__MODULE__, &(&1))
Agent.stop(__MODULE__)
MapSet.to_list(result)
end
defp start_link do
Agent.start_link(fn -> MapSet.new end, name: __MODULE__)
end
defp make_grid(h, w) do
for i <- 0..h-1, j <- 0..w-1, into: %{}, do: {{i,j}, true}
end
defp walk(h, w, grid, x, y, limit, cut, select) do
grid2 = grid |> Map.put({x,y}, false) |> Map.put({h-x-1,w-y-1}, false)
select2 = [{x,y} | select] |> Enum.sort
unless cut[select2] do
if length(select2) == limit do
Agent.update(__MODULE__, fn set -> MapSet.put(set, select2) end)
else
cut2 = Map.put(cut, select2, true)
search_next(grid2, select2)
|> Enum.each(fn {i,j} -> walk(h, w, grid2, i, j, limit, cut2, select2) end)
end
end
end
defp dirs(x, y), do: [{x+1, y}, {x-1, y}, {x, y-1}, {x, y+1}]
defp search_next(grid, select) do
(for {x,y} <- select, {i,j} <- dirs(x,y), grid[{i,j}], do: {i,j})
|> Enum.uniq
end
defp display(h, w) do
Agent.get(__MODULE__, &(&1))
|> Enum.each(fn select ->
grid = Enum.reduce(select, make_grid(h,w), fn {x,y},grid ->
%{grid | {x,y} => false}
end)
IO.puts to_string(h, w, grid)
end)
end
defp to_string(h, w, grid) do
text = for x <- 0..h*2, into: %{}, do: {x, String.duplicate(" ", w*4+1)}
text = Enum.reduce(0..h, text, fn i,acc ->
Enum.reduce(0..w, acc, fn j,txt ->
to_s(txt, i, j, grid)
end)
end)
Enum.map_join(0..h*2, "\n", fn i -> text[i] end)
end
defp to_s(text, i, j, grid) do
text = if grid[{i,j}] != grid[{i-1,j}], do: replace(text, i*2, j*4+1, "---"), else: text
text = if grid[{i,j}] != grid[{i,j-1}], do: replace(text, i*2+1, j*4, "|"), else: text
replace(text, i*2, j*4, "+")
end
defp replace(text, x, y, replacement) do
len = String.length(replacement)
Map.update!(text, x, fn str ->
String.slice(str, 0, y) <> replacement <> String.slice(str, y+len..-1)
end)
end
end
Rectangle.cut(2, 2) |> length |> IO.puts
Rectangle.cut(3, 4) |> length |> IO.puts
{{out}}
+---+---+
| |
+---+---+
| |
+---+---+
+---+---+
| | |
+ + +
| | |
+---+---+
2
+---+---+---+---+
| |
+ + +---+---+
| | |
+---+---+ + +
| |
+---+---+---+---+
+---+---+---+---+
| |
+ +---+ +---+
| | | | |
+---+ +---+ +
| |
+---+---+---+---+
+---+---+---+---+
| |
+---+ +---+ +
| | | | |
+ +---+ +---+
| |
+---+---+---+---+
+---+---+---+---+
| |
+---+---+ + +
| | |
+ + +---+---+
| |
+---+---+---+---+
+---+---+---+---+
| | |
+ + +---+ +
| | |
+ +---+ + +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ +---+ + +
| | | | |
+ + +---+ +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ + + + +
| | |
+ + + + +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ +---+ + +
| | |
+ + +---+ +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ + +---+ +
| | | | |
+ +---+ + +
| | |
+---+---+---+---+
9
```
## Go
{{trans|C}}
```go
package main
import "fmt"
var grid []byte
var w, h, last int
var cnt int
var next [4]int
var dir = [4][2]int{{0, -1}, {-1, 0}, {0, 1}, {1, 0}}
func walk(y, x int) {
if y == 0 || y == h || x == 0 || x == w {
cnt += 2
return
}
t := y*(w+1) + x
grid[t]++
grid[last-t]++
for i, d := range dir {
if grid[t+next[i]] == 0 {
walk(y+d[0], x+d[1])
}
}
grid[t]--
grid[last-t]--
}
func solve(hh, ww, recur int) int {
h = hh
w = ww
if h&1 != 0 {
h, w = w, h
}
switch {
case h&1 == 1:
return 0
case w == 1:
return 1
case w == 2:
return h
case h == 2:
return w
}
cy := h / 2
cx := w / 2
grid = make([]byte, (h+1)*(w+1))
last = len(grid) - 1
next[0] = -1
next[1] = -w - 1
next[2] = 1
next[3] = w + 1
if recur != 0 {
cnt = 0
}
for x := cx + 1; x < w; x++ {
t := cy*(w+1) + x
grid[t] = 1
grid[last-t] = 1
walk(cy-1, x)
}
cnt++
if h == w {
cnt *= 2
} else if w&1 == 0 && recur != 0 {
solve(w, h, 0)
}
return cnt
}
func main() {
for y := 1; y <= 10; y++ {
for x := 1; x <= y; x++ {
if x&1 == 0 || y&1 == 0 {
fmt.Printf("%d x %d: %d\n", y, x, solve(y, x, 1))
}
}
}
}
```
{{out}}
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
10 x 1: 1
10 x 2: 10
10 x 3: 115
10 x 4: 1228
10 x 5: 10295
10 x 6: 118276
10 x 7: 1026005
10 x 8: 11736888
10 x 9: 99953769
10 x 10: 1124140214
```
## Haskell
{{trans|Python}}
Calculation of the cuts happens in the ST monad, using a mutable STVector and a mutable STRef. The program style is therefore very imperative.
The strictness annotations in the Env type are necessary; otherwise, unevaluated thunks of updates of "env" would pile up with each recursion, ending in a stack overflow.
```Haskell
import qualified Data.Vector.Unboxed.Mutable as V
import Data.STRef
import Control.Monad (forM_, when)
import Control.Monad.ST
dir :: [(Int, Int)]
dir = [(1, 0), (-1, 0), (0, -1), (0, 1)]
data Env = Env { w, h, len, count, ret :: !Int, next :: ![Int] }
cutIt :: STRef s Env -> ST s ()
cutIt env = do
e <- readSTRef env
when (odd $ h e) $ modifySTRef env $ \en -> en { h = w e,
w = h e }
e <- readSTRef env
if odd (h e)
then modifySTRef env $ \en -> en { ret = 0 }
else
if w e == 1
then modifySTRef env $ \en -> en { ret = 1 }
else do
let blen = (h e + 1) * (w e + 1) - 1
t = (h e `div` 2) * (w e + 1) + (w e `div` 2)
modifySTRef env $ \en -> en { len = blen,
count = 0,
next = [ w e + 1, (negate $ w e) - 1, -1, 1] }
grid <- V.replicate (blen + 1) False
case odd (w e) of
True -> do
V.write grid t True
V.write grid (t + 1) True
walk grid (h e `div` 2) (w e `div` 2 - 1)
e1 <- readSTRef env
let res1 = count e1
modifySTRef env $ \en -> en { count = 0 }
walk grid (h e `div` 2 - 1) (w e `div` 2)
modifySTRef env $ \en -> en { ret = res1 +
(count en * 2) }
False -> do
V.write grid t True
walk grid (h e `div` 2) (w e `div` 2 - 1)
e2 <- readSTRef env
let count2 = count e2
if h e == w e
then modifySTRef env $ \en -> en { ret =
count2 * 2 }
else do
walk grid (h e `div` 2 - 1)
(w e `div` 2)
modifySTRef env $ \en -> en { ret =
count en }
where
walk grid y x = do
e <- readSTRef env
if y <= 0 || y >= h e || x <= 0 || x >= w e
then modifySTRef env $ \en -> en { count = count en + 1 }
else do
let t = y * (w e + 1) + x
V.write grid t True
V.write grid (len e - t) True
forM_ (zip (next e) [0..3]) $ \(n, d) -> do
g <- V.read grid (t + n)
when (not g) $
walk grid (y + fst (dir !! d)) (x + snd (dir !! d))
V.write grid t False
V.write grid (len e - t) False
cut :: (Int, Int) -> Int
cut (x, y) = runST $ do
env <- newSTRef $ Env { w = y, h = x, len = 0, count = 0, ret = 0, next = [] }
cutIt env
result <- readSTRef env
return $ ret result
main :: IO ()
main = do
mapM_ (\(x, y) -> when (even (x * y)) (putStrLn $
show x ++ " x " ++ show y ++ ": " ++ show (cut (x, y))))
[ (x, y) | x <- [1..10], y <- [1..x] ]
```
With GHC -O3 the run-time is about 39 times the D entry.
## J
```j
init=: - {. 1: NB. initial state: 1 square choosen
prop=: < {:,~2 ~:/\ ] NB. propagate: neighboring squares (vertically)
poss=: I.@,@(prop +. prop"1 +. prop&.|. +. prop&.|."1)
keep=: poss -. <:@#@, - I.@, NB. symmetrically valid possibilities
N=: <:@-:@#@, NB. how many neighbors to add
step=: [: ~.@; <@(((= i.@$) +. ])"0 _~ keep)"2
all=: step^:N@init
```
In other words, starting with a boolean matrix with one true square in one corner, make a list of all false squares which neighbor a true square, and then make each of those neighbors true, independently (discarding duplicate matrices from the resulting sequence of boolean matrices), and repeat this N times where N is (total cells divided by two)-1. Then discard those matrices where inverting them (boolean not), then flipping on horizontal and vertical axis is not an identity.
(In other words, this implementation uses a breadth first search -- breadth first searches tend to be natural in J because of the parallelism they offer.)
Example use:
```j
'.#' <"2@:{~ all 3 4
┌────┬────┬────┬────┬────┬────┬────┬────┬────┐
│.###│.###│..##│...#│...#│....│....│....│....│
│.#.#│..##│..##│..##│.#.#│..##│.#.#│#.#.│##..│
│...#│...#│..##│.###│.###│####│####│####│####│
└────┴────┴────┴────┴────┴────┴────┴────┴────┘
$ all 4 5
39 4 5
3 13$ '.#' <"2@:{~ all 4 5
┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
│.####│.####│.####│.####│.####│.####│..###│..###│..###│..###│..###│...##│...##│
│.####│.##.#│.#..#│..###│...##│....#│.####│.##.#│..###│...##│....#│.####│..###│
│....#│.#..#│.##.#│...##│..###│.####│....#│.#..#│...##│..###│.####│....#│...##│
│....#│....#│....#│....#│....#│....#│...##│...##│...##│...##│...##│..###│..###│
├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤
│...##│...##│...##│....#│....#│....#│....#│....#│....#│.....│.....│.....│.....│
│...##│....#│.#..#│.####│..###│...##│....#│.#..#│.##.#│.####│..###│...##│....#│
│..###│.####│.##.#│....#│...##│..###│.####│.##.#│.#..#│....#│...##│..###│.####│
│..###│..###│..###│.####│.####│.####│.####│.####│.####│#####│#####│#####│#####│
├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤
│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│
│.#..#│.##.#│..##.│...#.│.....│.#...│.##..│#.##.│#..#.│#....│##...│###..│####.│
│.##.#│.#..#│#..##│#.###│#####│###.#│##..#│#..#.│#.##.│####.│###..│##...│#....│
│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│
└─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘
```
## Java
{{works with|Java|7}}
```java
import java.util.*;
public class CutRectangle {
private static int[][] dirs = {{0, -1}, {-1, 0}, {0, 1}, {1, 0}};
public static void main(String[] args) {
cutRectangle(2, 2);
cutRectangle(4, 3);
}
static void cutRectangle(int w, int h) {
if (w % 2 == 1 && h % 2 == 1)
return;
int[][] grid = new int[h][w];
Stack stack = new Stack<>();
int half = (w * h) / 2;
long bits = (long) Math.pow(2, half) - 1;
for (; bits > 0; bits -= 2) {
for (int i = 0; i < half; i++) {
int r = i / w;
int c = i % w;
grid[r][c] = (bits & (1 << i)) != 0 ? 1 : 0;
grid[h - r - 1][w - c - 1] = 1 - grid[r][c];
}
stack.push(0);
grid[0][0] = 2;
int count = 1;
while (!stack.empty()) {
int pos = stack.pop();
int r = pos / w;
int c = pos % w;
for (int[] dir : dirs) {
int nextR = r + dir[0];
int nextC = c + dir[1];
if (nextR >= 0 && nextR < h && nextC >= 0 && nextC < w) {
if (grid[nextR][nextC] == 1) {
stack.push(nextR * w + nextC);
grid[nextR][nextC] = 2;
count++;
}
}
}
}
if (count == half) {
printResult(grid);
}
}
}
static void printResult(int[][] arr) {
for (int[] a : arr)
System.out.println(Arrays.toString(a));
System.out.println();
}
}
```
```txt
[2, 2]
[0, 0]
[2, 0]
[2, 0]
[2, 2, 2, 2]
[2, 2, 0, 0]
[0, 0, 0, 0]
[2, 2, 2, 0]
[2, 2, 0, 0]
[2, 0, 0, 0]
[2, 2, 0, 0]
[2, 2, 0, 0]
[2, 2, 0, 0]
[2, 0, 0, 0]
[2, 2, 0, 0]
[2, 2, 2, 0]
[2, 2, 2, 2]
[0, 2, 0, 2]
[0, 0, 0, 0]
[2, 2, 2, 2]
[2, 0, 2, 0]
[0, 0, 0, 0]
[2, 2, 2, 0]
[2, 0, 2, 0]
[2, 0, 0, 0]
[2, 0, 0, 0]
[2, 0, 2, 0]
[2, 2, 2, 0]
[2, 2, 2, 2]
[0, 0, 2, 2]
[0, 0, 0, 0]
```
## Julia
{{trans|C}}
```julia
const count = [0]
const dir = [[0, -1], [-1, 0], [0, 1], [1, 0]]
function walk(y, x, h, w, grid, len, next)
if y == 0 || y == h || x == 0 || x == w
count[1] += 2
return
end
t = y * (w + 1) + x
grid[t + 1] += UInt8(1)
grid[len - t + 1] += UInt8(1)
for i in 1:4
if grid[t + next[i] + 1] == 0
walk(y + dir[i][1], x + dir[i][2], h, w, grid, len, next)
end
end
grid[t + 1] -= 1
grid[len - t + 1] -= 1
end
function cutrectangle(hh, ww, recur)
if isodd(hh)
h, w = ww, hh
else
h, w = hh, ww
end
if isodd(h)
return 0
elseif w == 1
return 1
elseif w == 2
return h
elseif h == 2
return w
end
cy = div(h, 2)
cx = div(w, 2)
len = (h + 1) * (w + 1)
grid = zeros(UInt8, len)
len -= 1
next = [-1, -w - 1, 1, w + 1]
if recur
count[1] = 0
end
for x in cx + 1:w - 1
t = cy * (w + 1) + x
grid[t + 1] = 1
grid[len - t + 1] = 1
walk(cy - 1, x, h, w, grid, len, next)
end
count[1] += 1
if h == w
count[1] *= 2
elseif iseven(w) && recur
cutrectangle(w, h, false)
end
return count[1]
end
function runtest()
for y in 1:10, x in 1:y
if iseven(x) || iseven(y)
println("$y x $x: $(cutrectangle(y, x, true))")
end
end
end
runtest()
```
{{output}}
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
10 x 1: 1
10 x 2: 10
10 x 3: 115
10 x 4: 1228
10 x 5: 10295
10 x 6: 118276
10 x 7: 1026005
10 x 8: 11736888
10 x 9: 99953769
10 x 10: 1124140214
```
## Kotlin
{{trans|C}}
```scala
// version 1.0.6
object RectangleCutter {
private var w: Int = 0
private var h: Int = 0
private var len: Int = 0
private var cnt: Long = 0
private lateinit var grid: ByteArray
private val next = IntArray(4)
private val dir = arrayOf(
intArrayOf(0, -1),
intArrayOf(-1, 0),
intArrayOf(0, 1),
intArrayOf(1, 0)
)
private fun walk(y: Int, x: Int) {
if (y == 0 || y == h || x == 0 || x == w) {
cnt += 2
return
}
val t = y * (w + 1) + x
grid[t]++
grid[len - t]++
(0..3).filter { grid[t + next[it]] == 0.toByte() }
.forEach { walk(y + dir[it][0], x + dir[it][1]) }
grid[t]--
grid[len - t]--
}
fun solve(hh: Int, ww: Int, recur: Boolean): Long {
var t: Int
h = hh
w = ww
if ((h and 1) != 0) {
t = w
w = h
h = t
}
if ((h and 1) != 0) return 0L
if (w == 1) return 1L
if (w == 2) return h.toLong()
if (h == 2) return w.toLong()
val cy = h / 2
val cx = w / 2
len = (h + 1) * (w + 1)
grid = ByteArray(len)
len--
next[0] = -1
next[1] = -w - 1
next[2] = 1
next[3] = w + 1
if (recur) cnt = 0L
for (x in cx + 1 until w) {
t = cy * (w + 1) + x
grid[t] = 1
grid[len - t] = 1
walk(cy - 1, x)
}
cnt++
if (h == w) cnt *= 2
else if ((w and 1) == 0 && recur) solve(w, h, false)
return cnt
}
}
fun main(args: Array) {
for (y in 1..10) {
for (x in 1..y) {
if ((x and 1) == 0 || (y and 1) == 0) {
println("${"%2d".format(y)} x ${"%2d".format(x)}: ${RectangleCutter.solve(y, x, true)}")
}
}
}
}
```
{{out}}
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
10 x 1: 1
10 x 2: 10
10 x 3: 115
10 x 4: 1228
10 x 5: 10295
10 x 6: 118276
10 x 7: 1026005
10 x 8: 11736888
10 x 9: 99953769
10 x 10: 1124140214
```
## Perl
{{trans|C}}
Output is identical to C's.
```perl
use strict;
use warnings;
my @grid = 0;
my ($w, $h, $len);
my $cnt = 0;
my @next;
my @dir = ([0, -1], [-1, 0], [0, 1], [1, 0]);
sub walk {
my ($y, $x) = @_;
if (!$y || $y == $h || !$x || $x == $w) {
$cnt += 2;
return;
}
my $t = $y * ($w + 1) + $x;
$grid[$_]++ for $t, $len - $t;
for my $i (0 .. 3) {
if (!$grid[$t + $next[$i]]) {
walk($y + $dir[$i]->[0], $x + $dir[$i]->[1]);
}
}
$grid[$_]-- for $t, $len - $t;
}
sub solve {
my ($hh, $ww, $recur) = @_;
my ($t, $cx, $cy, $x);
($h, $w) = ($hh, $ww);
if ($h & 1) { ($t, $w, $h) = ($w, $h, $w); }
if ($h & 1) { return 0; }
if ($w == 1) { return 1; }
if ($w == 2) { return $h; }
if ($h == 2) { return $w; }
{
use integer;
($cy, $cx) = ($h / 2, $w / 2);
}
$len = ($h + 1) * ($w + 1);
@grid = ();
$grid[$len--] = 0;
@next = (-1, -$w - 1, 1, $w + 1);
if ($recur) { $cnt = 0; }
for ($x = $cx + 1; $x < $w; $x++) {
$t = $cy * ($w + 1) + $x;
@grid[$t, $len - $t] = (1, 1);
walk($cy - 1, $x);
}
$cnt++;
if ($h == $w) {
$cnt *= 2;
} elsif (!($w & 1) && $recur) {
solve($w, $h);
}
return $cnt;
}
sub MAIN {
print "ok\n";
my ($y, $x);
for my $y (1 .. 10) {
for my $x (1 .. $y) {
if (!($x & 1) || !($y & 1)) {
printf("%d x %d: %d\n", $y, $x, solve($y, $x, 1));
}
}
}
}
MAIN();
```
## Perl 6
{{trans|C}}
This is a very dumb, straightforward translation of the C code. It is very slow so we'll interrupt the execution and display the partial output.
```perl6
subset Byte of Int where ^256;
my @grid of Byte = 0;
my Int ($w, $h, $len);
my Int $cnt = 0;
my @next;
my @dir = [0, -1], [-1, 0], [0, 1], [1, 0];
sub walk(Int $y, Int $x) {
my ($i, $t);
if !$y || $y == $h || !$x || $x == $w {
$cnt += 2;
return;
}
$t = $y * ($w + 1) + $x;
@grid[$t]++, @grid[$len - $t]++;
loop ($i = 0; $i < 4; $i++) {
if !@grid[$t + @next[$i]] {
walk($y + @dir[$i][0], $x + @dir[$i][1]);
}
}
@grid[$t]--, @grid[$len - $t]--;
}
sub solve(Int $hh, Int $ww, Int $recur) returns Int {
my ($t, $cx, $cy, $x);
$h = $hh, $w = $ww;
if $h +& 1 { $t = $w, $w = $h, $h = $t; }
if $h +& 1 { return 0; }
if $w == 1 { return 1; }
if $w == 2 { return $h; }
if $h == 2 { return $w; }
$cy = $h div 2, $cx = $w div 2;
$len = ($h + 1) * ($w + 1);
@grid = ();
@grid[$len--] = 0;
@next[0] = -1;
@next[1] = -$w - 1;
@next[2] = 1;
@next[3] = $w + 1;
if $recur { $cnt = 0; }
loop ($x = $cx + 1; $x < $w; $x++) {
$t = $cy * ($w + 1) + $x;
@grid[$t] = 1;
@grid[$len - $t] = 1;
walk($cy - 1, $x);
}
$cnt++;
if $h == $w {
$cnt *= 2;
} elsif !($w +& 1) && $recur {
solve($w, $h, 0);
}
return $cnt;
}
my ($y, $x);
loop ($y = 1; $y <= 9; $y++) {
loop ($x = 1; $x <= $y; $x++) {
if (!($x +& 1) || !($y +& 1)) {
printf("%d x %d: %d\n", $y, $x, solve($y, $x, 1));
}
}
}
```
{{out}}
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
^C
```
## Phix
Using a completely different home-brewed algorithm, slightly sub-optimal as noted in the code.
```Phix
integer show = 2, -- max number to show
-- (nb mirrors are not shown)
chance = 1000 -- 1=always, 2=50%, 3=33%, etc
sequence grid
integer gh, -- = length(grid),
gw -- = length(grid[1])
integer ty1, ty2, tx1, tx2 -- target {y,x}s
procedure mirror(integer y, x, ch)
-- plant/reset ch and the symmetric copy
grid[y,x] = ch
grid[gh-y+1,gw-x+1] = ch
end procedure
enum RIGHT, UP, DOWN, LEFT
constant dyx = {{0,+1},{-1,0},{+1,0},{0,-1}},
chx = "-||-"
function search(integer y, x, d, level)
integer count = 0
if level=0 or grid[y,x]!='x' then
mirror(y,x,'x')
integer {dy,dx} = dyx[d],
{ny,nx} = {y+dy,x+dx},
{yy,xx} = {y+dy*2,x+dx*3}
if grid[ny,nx]=' ' then
integer c = chx[d]
mirror(ny,nx,c)
if c='-' then
mirror(ny,nx+dx,c)
end if
integer meet = (yy=ty1 or yy=ty2) and (xx=tx1 or xx=tx2)
if meet then
count = 1
if show and rand(chance)=chance then
show -= 1
sequence g = grid -- (make copy/avoid reset)
-- fill in(/overwrite) the last cut, if any
if ty1!=ty2 then g[ty1+1,tx1] = '|'
elsif tx1!=tx2 then g[ty1][tx1+1..tx1+2] = "--"
end if
puts(1,join(g,'\n')&"\n\n")
end if
else
if grid[yy,xx]='+' then -- (minor gain)
for d=RIGHT to LEFT do -- (kinda true!)
count += search(yy,xx,d,level+1)
end for
end if
end if
mirror(ny,nx,' ')
if c='-' then
mirror(ny,nx+dx,' ')
end if
end if
if level!=0 then
-- ((level=0)==leave outer edges 'x' for next iteration)
mirror(y,x,'+')
end if
end if
return count
end function
function odd(integer n) return remainder(n,2)=1 end function
function even(integer n) return remainder(n,2)=0 end function
procedure make_grid(integer w,h)
-- The outer edges are 'x'; the inner '+' become 'x' when visited.
-- Likewise edges are cuts but the inner ones get filled in later.
sequence tb = join(repeat("x",w+1),"--"),
hz = join('x'&repeat("+",w-1)&'x'," ")&"\n",
vt = "|"&repeat(' ',w*3-1)&"|\n"
grid = split(tb&"\n"&join(repeat(vt,h),hz)&tb,'\n')
-- set size (for mirroring) and target info:
gh = length(grid) gw = length(grid[1])
ty1 = h+even(h) ty2 = ty1+odd(h)*2
tx1 = floor(w/2)*3+1 tx2 = tx1+odd(w)*3
end procedure
function side(integer w, h)
make_grid(w,h)
-- search top to mid-point
integer count = 0, last = 0
for r=3 to h+1 by 2 do
last = search(r,1,RIGHT,0) -- left to right
count += 2*last
end for
if even(h) then
count -= last -- (un-double the centre line)
end if
return count
end function
--atom t0 = time()
-- nb sub-optimal: obviously "grid" was designed for easy display, rather than speed.
for y=1 to 9 do -- 24s
--for y=1 to 10 do -- (gave up on >10x8)
for x=1 to y do
-- for x=1 to min(y,8) do -- 4 mins 16s (with y to 10)
if even(x*y) then
integer count = side(x,y)
if x=y then
count *= 2
else
count += side(y,x)
end if
printf(1,"%d x %d: %d\n", {y, x, count})
end if
end for
end for
--?elapsed(time()-t0)
```
{{out}}
Includes two random grids
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
x--x--x--x--x--x--x
| |
x--x + + + + x
| | |
x x x--x--x + x
| | | | |
x x--x x--x + x
| | |
x + x--x x--x x
| | | | |
x + x--x--x x x
| | |
x + + + + x--x
| |
x--x--x--x--x--x--x
x--x--x--x--x--x--x--x
| |
x + x--x--x--x--x x
| | | |
x--x--x x--x--x x x
| | | | |
x x--x x--x x--x x
| | | | |
x x x--x--x x--x--x
| | | |
x x--x--x--x--x + x
| |
x--x--x--x--x--x--x--x
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
10 x 1: 1
10 x 2: 10
10 x 3: 115
10 x 4: 1228
10 x 5: 10295
10 x 6: 118276
10 x 7: 1026005
10 x 8: 11736888
```
## Python
{{trans|D}}
```python
def cut_it(h, w):
dirs = ((1, 0), (-1, 0), (0, -1), (0, 1))
if h & 1: h, w = w, h
if h & 1: return 0
if w == 1: return 1
count = 0
next = [w + 1, -w - 1, -1, 1]
blen = (h + 1) * (w + 1) - 1
grid = [False] * (blen + 1)
def walk(y, x, count):
if not y or y == h or not x or x == w:
return count + 1
t = y * (w + 1) + x
grid[t] = grid[blen - t] = True
if not grid[t + next[0]]:
count = walk(y + dirs[0][0], x + dirs[0][1], count)
if not grid[t + next[1]]:
count = walk(y + dirs[1][0], x + dirs[1][1], count)
if not grid[t + next[2]]:
count = walk(y + dirs[2][0], x + dirs[2][1], count)
if not grid[t + next[3]]:
count = walk(y + dirs[3][0], x + dirs[3][1], count)
grid[t] = grid[blen - t] = False
return count
t = h // 2 * (w + 1) + w // 2
if w & 1:
grid[t] = grid[t + 1] = True
count = walk(h // 2, w // 2 - 1, count)
res = count
count = 0
count = walk(h // 2 - 1, w // 2, count)
return res + count * 2
else:
grid[t] = True
count = walk(h // 2, w // 2 - 1, count)
if h == w:
return count * 2
count = walk(h // 2 - 1, w // 2, count)
return count
def main():
for w in xrange(1, 10):
for h in xrange(1, w + 1):
if not((w * h) & 1):
print "%d x %d: %d" % (w, h, cut_it(w, h))
main()
```
Output:
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
```
### Faster version
{{trans|D}}
```python
try:
import psyco
except ImportError:
pass
else:
psyco.full()
w, h = 0, 0
count = 0
vis = []
def cwalk(y, x, d):
global vis, count, w, h
if not y or y == h or not x or x == w:
count += 1
return
vis[y][x] = vis[h - y][w - x] = 1
if x and not vis[y][x - 1]:
cwalk(y, x - 1, d | 1)
if (d & 1) and x < w and not vis[y][x+1]:
cwalk(y, x + 1, d|1)
if y and not vis[y - 1][x]:
cwalk(y - 1, x, d | 2)
if (d & 2) and y < h and not vis[y + 1][x]:
cwalk(y + 1, x, d | 2)
vis[y][x] = vis[h - y][w - x] = 0
def count_only(x, y):
global vis, count, w, h
count = 0
w = x
h = y
if (h * w) & 1:
return count
if h & 1:
w, h = h, w
vis = [[0] * (w + 1) for _ in xrange(h + 1)]
vis[h // 2][w // 2] = 1
if w & 1:
vis[h // 2][w // 2 + 1] = 1
res = 0
if w > 1:
cwalk(h // 2, w // 2 - 1, 1)
res = 2 * count - 1
count = 0
if w != h:
cwalk(h // 2 + 1, w // 2, 3 if (w & 1) else 2)
res += 2 * count - (not (w & 1))
else:
res = 1
if w == h:
res = 2 * res + 2
return res
def main():
for y in xrange(1, 10):
for x in xrange(1, y + 1):
if not (x & 1) or not (y & 1):
print "%d x %d: %d" % (y, x, count_only(x, y))
main()
```
The output is the same.
## Racket
```racket
#lang racket
(define (cuts W H [count 0]) ; count = #f => visualize instead
(define W1 (add1 W)) (define H1 (add1 H))
(define B (make-vector (* W1 H1) #f))
(define (fD d) (cadr (assq d '([U D] [D U] [L R] [R L] [#f #f] [#t #t]))))
(define (fP p) (- (* W1 H1) p 1))
(define (Bset! p d) (vector-set! B p d) (vector-set! B (fP p) (fD d)))
(define center (/ (fP 0) 2))
(when (integer? center) (Bset! center #t))
(define (run c* d)
(define p (- center c*))
(Bset! p d)
(let loop ([p p])
(define-values [q r] (quotient/remainder p W1))
(if (and (< 0 r W) (< 0 q H))
(for ([d '(U D L R)])
(define n (+ p (case d [(U) (- W1)] [(D) W1] [(L) -1] [(R) 1])))
(unless (vector-ref B n) (Bset! n (fD d)) (loop n) (Bset! n #f)))
(if count (set! count (add1 count)) (visualize B W H))))
(Bset! p #f))
(when (even? W) (run (if (odd? H) (/ W1 2) W1) 'D))
(when (even? H) (run (if (odd? W) 1/2 1) 'R))
(or count (void)))
(define (visualize B W H)
(define W2 (+ 2 (* W 2))) (define H2 (+ 1 (* H 2)))
(define str (make-string (* H2 W2) #\space))
(define (Sset! i c) (string-set! str i c))
(for ([i (in-range (- W2 1) (* W2 H2) W2)]) (Sset! i #\newline))
(for ([i (in-range 0 (- W2 1))]) (Sset! i #\#) (Sset! (+ i (* W2 H 2)) #\#))
(for ([i (in-range 0 (* W2 H2) W2)]) (Sset! i #\#) (Sset! (+ i W2 -2) #\#))
(for* ([i (add1 W)] [j (add1 H)])
(define p (* 2 (+ i (* j W2))))
(define b (vector-ref B (+ i (* j (+ W 1)))))
(cond [b (Sset! p #\#)
(define d (case b [(U) (- W2)] [(D) W2] [(R) 1] [(L) -1]))
(when (integer? d) (Sset! (+ p d) #\#))]
[(equal? #\space (string-ref str p)) (Sset! p #\.)]))
(display str) (newline))
(printf "Counts:\n")
(for* ([W (in-range 1 10)] [H (in-range 1 (add1 W))]
#:unless (and (odd? W) (odd? H)))
(printf "~s x ~s: ~s\n" W H (cuts W H)))
(newline)
(cuts 4 3 #f)
```
{{out}}
```txt
Counts:
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
#########
# # #
# . # . #
# # #
# . # . #
# # #
#########
#########
# # #
# ### . #
# # #
# . ### #
# # #
#########
#########
# # #
# ### # #
# # # # #
# # ### #
# # #
#########
#########
# #
# ### ###
# # # # #
### ### #
# #
#########
#########
# #
##### . #
# # #
# . #####
# #
#########
#########
# # #
# . ### #
# # #
# ### . #
# # #
#########
#########
# # #
# # ### #
# # # # #
# ### # #
# # #
#########
#########
# #
### ### #
# # # # #
# ### ###
# #
#########
#########
# #
# . #####
# # #
##### . #
# #
#########
```
## REXX
### idiomatic
```rexx
/*REXX program cuts rectangles into two symmetric pieces, the rectangles are cut along */
/*────────────────────────────────────────────────── unit dimensions and may be rotated.*/
numeric digits 20 /*be able to handle some big integers. */
parse arg N .; if N=='' | N=="," then N=10 /*N not specified? Then use default.*/
dir.=0; dir.0.1=-1; dir.1.0=-1; dir.2.1=1; dir.3.0=1 /*the four directions.*/
do y=2 to N; say /*calculate rectangles up to size NxN.*/
do x=1 for y; if x//2 & y//2 then iterate /*not if both X&Y odd.*/
z=solve(y,x,1); _=comma(z); _=right(_, max(14, length(_))) /*align the output. */
say right(y,9) "x" right(x,2) 'rectangle can be cut' _ "way"s(z).
end /*x*/
end /*y*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
comma: procedure; arg _; do k=length(_)-3 to 1 by -3; _=insert(',',_,k); end; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
s: if arg(1)=1 then return arg(3); return word(arg(2) 's', 1) /*pluralizer.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
solve: procedure expose # dir. @. h len next. w; @.=0 /*zero rectangle coördinates.*/
parse arg h,w,recur /*get values for some args. */
if h//2 then do; t=w; w=h; h=t; if h//2 then return 0
end
if w==1 then return 1
if w==2 then return h
if h==2 then return w /* [↓] % is REXX's integer division.*/
cy=h % 2; cx=w % 2; wp=w + 1 /*cut the [XY] rectangle in half. */
len=(h+1) * wp - 1 /*extend the area of the rectangle. */
next.0=-1; next.1=-wp; next.2=1; next.3=wp /*direction & distance.*/
if recur then #=0
do x=cx+1 to w-1; t=x + cy*wp; @.t=1; _=len - t; @._=1
call walk cy-1, x
end /*x*/
#=#+1
if h==w then #=# + # /*double the count of rectangle cuts. */
else if w//2==0 & recur then call solve w, h, 0
return #
/*──────────────────────────────────────────────────────────────────────────────────────*/
walk: procedure expose # dir. @. h len next. w wp; parse arg y,x
if y==h | x==0 | x==w | y==0 then do; #= #+2; return; end
t=x + y*wp; @.t=@.t + 1; _=len - t
@._=@._+1
do j=0 for 4; _=t + next.j /*try each of four directions.*/
if @._==0 then call walk y + dir.j.0, x + dir.j.1
end /*j*/
@.t=@.t - 1
_=len - t; @._=@._ - 1; return
```
{{out|output|text= when using the default input:}}
```txt
2 x 1 rectangle can be cut 1 way.
2 x 2 rectangle can be cut 2 ways.
3 x 2 rectangle can be cut 3 ways.
4 x 1 rectangle can be cut 1 way.
4 x 2 rectangle can be cut 4 ways.
4 x 3 rectangle can be cut 9 ways.
4 x 4 rectangle can be cut 22 ways.
5 x 2 rectangle can be cut 5 ways.
5 x 4 rectangle can be cut 39 ways.
6 x 1 rectangle can be cut 1 way.
6 x 2 rectangle can be cut 6 ways.
6 x 3 rectangle can be cut 23 ways.
6 x 4 rectangle can be cut 90 ways.
6 x 5 rectangle can be cut 263 ways.
6 x 6 rectangle can be cut 1,018 ways.
7 x 2 rectangle can be cut 7 ways.
7 x 4 rectangle can be cut 151 ways.
7 x 6 rectangle can be cut 2,947 ways.
8 x 1 rectangle can be cut 1 way.
8 x 2 rectangle can be cut 8 ways.
8 x 3 rectangle can be cut 53 ways.
8 x 4 rectangle can be cut 340 ways.
8 x 5 rectangle can be cut 1,675 ways.
8 x 6 rectangle can be cut 11,174 ways.
8 x 7 rectangle can be cut 55,939 ways.
8 x 8 rectangle can be cut 369,050 ways.
9 x 2 rectangle can be cut 9 ways.
9 x 4 rectangle can be cut 553 ways.
9 x 6 rectangle can be cut 31,721 ways.
9 x 8 rectangle can be cut 1,812,667 ways.
10 x 1 rectangle can be cut 1 way.
10 x 2 rectangle can be cut 10 ways.
10 x 3 rectangle can be cut 115 ways.
10 x 4 rectangle can be cut 1,228 ways.
10 x 5 rectangle can be cut 10,295 ways.
10 x 6 rectangle can be cut 118,276 ways.
10 x 7 rectangle can be cut 1,026,005 ways.
10 x 8 rectangle can be cut 11,736,888 ways.
10 x 9 rectangle can be cut 99,953,769 ways.
10 x 10 rectangle can be cut 1,124,140,214 ways.
```
### optimized
This version replaced the (first) multiple clause '''if''' instructions in the '''walk''' subroutine with a
''short circuit'' version. Other optimizations were also made. This made the program about 20% faster.
A test run was executed to determine the order of the '''if''' statements (by counting which
comparison would yield the most benefit by placing it first).
```rexx
/*REXX program cuts rectangles into two symmetric pieces, the rectangles are cut along */
/*────────────────────────────────────────────────── unit dimensions and may be rotated.*/
numeric digits 20 /*be able to handle some big integers. */
parse arg N .; if N=='' | N=="," then N=10 /*N not specified? Then use default.*/
dir.=0; dir.0.1= -1; dir.1.0= -1; dir.2.1= 1; dir.3.0= 1 /*the 4 directions.*/
do y=2 to N; yEven= y//2; say /*calculate rectangles up to size NxN.*/
do x=1 for y; if x//2 then if yEven then iterate /*not if both X&Y odd*/
z= solve(y,x,1); _=comma(z); _=right(_, max(14, length(_))) /*align the output. */
say right(y, 9) "x" right(x, 2) 'rectangle can be cut' _ "way"s(z).
end /*x*/
end /*y*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
comma: procedure; arg _; do k=length(_)-3 to 1 by -3; _=insert(',',_,k); end; return _
s: if arg(1)=1 then return arg(3); return word(arg(2) 's', 1) /*pluralizer.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
solve: procedure expose # dir. @. h len next. w; @.=0 /*zero rectangle coördinates.*/
parse arg h,w,recur /*get values for some args. */
if h//2 then do; t= w; w= h; h= t; if h//2 then return 0
end
if w==1 then return 1
if w==2 then return h
if h==2 then return w /* [↓] % is REXX's integer division.*/
cy= h % 2; cx= w % 2; wp= w + 1 /*cut the [XY] rectangle in half. */
len= (h+1) * wp - 1 /*extend the area of the rectangle. */
next.0= -1; next.1= -wp; next.2= 1; next.3= wp /*direction & distance.*/
if recur then #= 0
do x=cx+1 to w-1; t= x + cy*wp; @.t= 1; _= len - t; @._= 1
call walk cy-1, x
end /*x*/
#= #+1
if h==w then #= # + # /*double the count of rectangle cuts. */
else if w//2==0 then if recur then call solve w, h, 0
return #
/*──────────────────────────────────────────────────────────────────────────────────────*/
walk: procedure expose # dir. @. h len next. w wp; parse arg y,x
if y==h then do; #= #+2; return; end /* ◄──┐ REXX short circuit. */
if x==0 then do; #= #+2; return; end /* ◄──┤ " " " */
if x==w then do; #= #+2; return; end /* ◄──┤ " " " */
if y==0 then do; #= #+2; return; end /* ◄──┤ " " " */
t= x + y*wp; @.t= @.t + 1; _= len - t /* │ordered by most likely ►──┐*/
@._= @._+1 /* └──────────────────────────┘*/
do j=0 for 4; _= t + next.j /*try each of the four directions.*/
if @._==0 then do; yn= y + dir.j.0; xn= x + dir.j.1
if yn==h then do; #= #+2; iterate; end
if xn==0 then do; #= #+2; iterate; end
if xn==w then do; #= #+2; iterate; end
if yn==0 then do; #= #+2; iterate; end
call walk yn, xn
end
end /*j*/
@.t= @.t - 1
_= len - t; @._= @._ - 1; return
```
{{out|output|text= is the same as the idiomatic version (above).}}
## Ruby
{{trans|Python}}
```ruby
def cut_it(h, w)
if h.odd?
return 0 if w.odd?
h, w = w, h
end
return 1 if w == 1
nxt = [[w+1, 1, 0], [-w-1, -1, 0], [-1, 0, -1], [1, 0, 1]] # [next,dy,dx]
blen = (h + 1) * (w + 1) - 1
grid = [false] * (blen + 1)
walk = lambda do |y, x, count=0|
return count+1 if y==0 or y==h or x==0 or x==w
t = y * (w + 1) + x
grid[t] = grid[blen - t] = true
nxt.each do |nt, dy, dx|
count += walk[y + dy, x + dx] unless grid[t + nt]
end
grid[t] = grid[blen - t] = false
count
end
t = h / 2 * (w + 1) + w / 2
if w.odd?
grid[t] = grid[t + 1] = true
count = walk[h / 2, w / 2 - 1]
count + walk[h / 2 - 1, w / 2] * 2
else
grid[t] = true
count = walk[h / 2, w / 2 - 1]
return count * 2 if h == w
count + walk[h / 2 - 1, w / 2]
end
end
for w in 1..9
for h in 1..w
puts "%d x %d: %d" % [w, h, cut_it(w, h)] if (w * h).even?
end
end
```
{{out}}
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 1
6 x 2: 6
6 x 3: 23
6 x 4: 90
6 x 5: 263
6 x 6: 1018
7 x 2: 7
7 x 4: 151
7 x 6: 2947
8 x 1: 1
8 x 2: 8
8 x 3: 53
8 x 4: 340
8 x 5: 1675
8 x 6: 11174
8 x 7: 55939
8 x 8: 369050
9 x 2: 9
9 x 4: 553
9 x 6: 31721
9 x 8: 1812667
```
### Show each of the cuts
```ruby
class Rectangle
DIRS = [[1, 0], [-1, 0], [0, -1], [0, 1]]
def initialize(h, w)
raise ArgumentError if (h.odd? and w.odd?) or h<=0 or w<=0
@h, @w = h, w
@limit = h * w / 2
end
def cut(disp=true)
@cut = {}
@select = []
@result = []
@grid = make_grid
walk(0,0)
display if disp
@result
end
def make_grid
Array.new(@h+1) {|i| Array.new(@w+1) {|j| true if i<@h and j<@w }}
end
def walk(y, x)
@grid[y][x] = @grid[@h-y-1][@w-x-1] = false
@select.push([y,x])
select = @select.sort
unless @cut[select]
@cut[select] = true
if @select.size == @limit
@result << select
else
search_next.each {|yy,xx| walk(yy,xx)}
end
end
@select.pop
@grid[y][x] = @grid[@h-y-1][@w-x-1] = true
end
def search_next
nxt = {}
@select.each do |y,x|
DIRS.each do |dy, dx|
nxt[[y+dy, x+dx]] = true if @grid[y+dy][x+dx]
end
end
nxt.keys
end
def display
@result.each do |select|
@grid = make_grid
select.each {|y,x| @grid[y][x] = false}
puts to_s
end
end
def to_s
text = Array.new(@h*2+1) {" " * (@w*4+1)}
for i in 0..@h
for j in 0..@w
text[i*2][j*4+1,3] = "---" if @grid[i][j] != @grid[i-1][j]
text[i*2+1][j*4] = "|" if @grid[i][j] != @grid[i][j-1]
text[i*2][j*4] = "+"
end
end
text.join("\n")
end
end
rec = Rectangle.new(2,2)
puts rec.cut.size
rec = Rectangle.new(3,4)
puts rec.cut.size
```
{{out}}
+---+---+
| | |
+ + +
| | |
+---+---+
+---+---+
| |
+---+---+
| |
+---+---+
2
+---+---+---+---+
| | |
+ + +---+ +
| | |
+ +---+ + +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ + + + +
| | |
+ + + + +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ +---+ + +
| | | | |
+ + +---+ +
| | |
+---+---+---+---+
+---+---+---+---+
| |
+ + +---+---+
| | |
+---+---+ + +
| |
+---+---+---+---+
+---+---+---+---+
| |
+ +---+ +---+
| | | | |
+---+ +---+ +
| |
+---+---+---+---+
+---+---+---+---+
| | |
+ +---+ + +
| | |
+ + +---+ +
| | |
+---+---+---+---+
+---+---+---+---+
| | |
+ + +---+ +
| | | | |
+ +---+ + +
| | |
+---+---+---+---+
+---+---+---+---+
| |
+---+ +---+ +
| | | | |
+ +---+ +---+
| |
+---+---+---+---+
+---+---+---+---+
| |
+---+---+ + +
| | |
+ + +---+---+
| |
+---+---+---+---+
9
```
## Rust
{{trans|Python}}
```rust
fn cwalk(mut vis: &mut Vec>, count: &mut isize, w: usize, h: usize, y: usize, x: usize, d: usize) {
if x == 0 || y == 0 || x == w || y == h {
*count += 1;
return;
}
vis[y][x] = true;
vis[h - y][w - x] = true;
if x != 0 && ! vis[y][x - 1] {
cwalk(&mut vis, count, w, h, y, x - 1, d | 1);
}
if d & 1 != 0 && x < w && ! vis[y][x+1] {
cwalk(&mut vis, count, w, h, y, x + 1, d | 1);
}
if y != 0 && ! vis[y - 1][x] {
cwalk(&mut vis, count, w, h, y - 1, x, d | 2);
}
if d & 2 != 0 && y < h && ! vis[y + 1][x] {
cwalk(&mut vis, count, w, h, y + 1, x, d | 2);
}
vis[y][x] = false;
vis[h - y][w - x] = false;
}
fn count_only(x: usize, y: usize) -> isize {
let mut count = 0;
let mut w = x;
let mut h = y;
if (h * w) & 1 != 0 {
return count;
}
if h & 1 != 0 {
std::mem::swap(&mut w, &mut h);
}
let mut vis = vec![vec![false; w + 1]; h + 1];
vis[h / 2][w / 2] = true;
if w & 1 != 0 {
vis[h / 2][w / 2 + 1] = true;
}
let mut res;
if w > 1 {
cwalk(&mut vis, &mut count, w, h, h / 2, w / 2 - 1, 1);
res = 2 * count - 1;
count = 0;
if w != h {
cwalk(&mut vis, &mut count, w, h, h / 2 + 1, w / 2, if w & 1 != 0 { 3 } else { 2 });
}
res += 2 * count - if w & 1 == 0 { 1 } else { 0 };
}
else {
res = 1;
}
if w == h {
res = 2 * res + 2;
}
res
}
fn main() {
for y in 1..10 {
for x in 1..y + 1 {
if x & 1 == 0 || y & 1 == 0 {
println!("{} x {}: {}", y, x, count_only(x, y));
}
}
}
}
```
## Tcl
{{trans|C}}
```tcl
package require Tcl 8.5
proc walk {y x} {
global w ww h cnt grid len
if {!$y || $y==$h || !$x || $x==$w} {
incr cnt 2
return
}
set t [expr {$y*$ww + $x}]
set m [expr {$len - $t}]
lset grid $t [expr {[lindex $grid $t] + 1}]
lset grid $m [expr {[lindex $grid $m] + 1}]
if {![lindex $grid [expr {$y*$ww + $x-1}]]} {
walk $y [expr {$x-1}]
}
if {![lindex $grid [expr {($y-1)*$ww + $x}]]} {
walk [expr {$y-1}] $x
}
if {![lindex $grid [expr {$y*$ww + $x+1}]]} {
walk $y [expr {$x+1}]
}
if {![lindex $grid [expr {($y+1)*$ww + $x}]]} {
walk [expr {$y+1}] $x
}
lset grid $t [expr {[lindex $grid $t] - 1}]
lset grid $m [expr {[lindex $grid $m] - 1}]
}
# Factored out core of [solve]
proc SolveCore {} {
global w ww h cnt grid len
set ww [expr {$w+1}]
set cy [expr {$h / 2}]
set cx [expr {$w / 2}]
set len [expr {($h+1) * $ww}]
set grid [lrepeat $len 0]
incr len -1
for {set x $cx;incr x} {$x < $w} {incr x} {
set t [expr {$cy*$ww+$x}]
lset grid $t 1
lset grid [expr {$len - $t}] 1
walk [expr {$cy - 1}] $x
}
incr cnt
}
proc solve {H W} {
global w h cnt
set h $H
set w $W
if {$h & 1} {
set h $W
set w $H
}
if {$h & 1} {
return 0
}
if {$w==1} {return 1}
if {$w==2} {return $h}
if {$h==2} {return $w}
set cnt 0
SolveCore
if {$h==$w} {
incr cnt $cnt
} elseif {!($w & 1)} {
lassign [list $w $h] h w
SolveCore
}
return $cnt
}
apply {{limit} {
for {set yy 1} {$yy <= $limit} {incr yy} {
for {set xx 1} {$xx <= $yy} {incr xx} {
if {!($xx&1 && $yy&1)} {
puts [format "%d x %d: %ld" $yy $xx [solve $yy $xx]]
}
}
}
}} 10
```
Output is identical.
## zkl
{{trans|Ruby}}
```zkl
fcn cut_it(h,w){
if(h.isOdd){
if(w.isOdd) return(0);
t,h,w=h,w,t; // swap w,h: a,b=c,d --> a=c; b=d; so need a tmp
}
if(w==1) return(1);
nxt :=T(T(w+1, 1,0), T(-w-1, -1,0), T(-1, 0,-1), T(1, 0,1)); #[next, dy,dx]
blen:=(h + 1)*(w + 1) - 1;
grid:=(blen + 1).pump(List(),False); //-->L(False,False...)
walk:='wrap(y,x){ // lambda closure
if(y==0 or y==h or x==0 or x==w) return(1);
count,t:=0,y*(w + 1) + x;
grid[t]=grid[blen - t]=True;
foreach nt,dy,dx in (nxt){
if(not grid[t + nt]) count+=self.fcn(y + dy, x + dx,vm.pasteArgs(2));
}
grid[t]=grid[blen - t]=False;
count
};
t:=h/2*(w + 1) + w/2;
if(w.isOdd){
grid[t]=grid[t + 1]=True;
count:=walk(h/2, w/2 - 1);
count + walk(h/2 - 1, w/2)*2;
}else{
grid[t]=True;
count:=walk(h/2, w/2 - 1);
if(h==w) return(count*2);
count + walk(h/2 - 1, w/2);
}
}
```
Note the funkiness in walk: vm.pasteArgs. This is because zkl functions are unaware of their scope, so a closure is needed (when calling walk) to capture state (nxt, blen, grid, h, w). Rather than creating a closure object each call, that state is passed in the arg list. So, when doing recursion, that state needs to be restored to the stack (the compiler isn't smart enough to recognize this case).
```zkl
foreach w,h in ([1..9],[1..w]){
if((w*h).isEven) println("%d x %d: %d".fmt(w, h, cut_it(w,h)));
}
```
{{out}}
Output is identical.
```txt
2 x 1: 1
2 x 2: 2
3 x 2: 3
4 x 1: 1
4 x 2: 4
4 x 3: 9
4 x 4: 22
...
9 x 2: 9
9 x 4: 553
9 x 6: 31721
```
{{omit from|GUISS}}
{{omit from|Lilypond}}
{{omit from|TPP}}
[[Category:Geometry]]