Cut a rectangle

From Rosetta Code
Task
Cut a rectangle
You are encouraged to solve this task according to the task description, using any language you may know.

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.

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[edit]

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
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

More awkward solution: after compiling, run ./a.out -v [width] [height] for display of cuts.

#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[edit]

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
2 x 1: 2
2 x 2: 2
3 x 2: 3
4 x 1: 4
4 x 2: 4
4 x 3: 9
4 x 4: 22
5 x 2: 5
5 x 4: 39
6 x 1: 6
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: 8
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

D[edit]

Translation of: C
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));
}
Output:
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[edit]

 
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
 
Output:
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[edit]

Translation of: Ruby

Count only[edit]

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)
Output:
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[edit]

Works with: Elixir version 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
Output:
+---+---+
|       |
+---+---+
|       |
+---+---+
+---+---+
|   |   |
+   +   +
|   |   |
+---+---+
2
+---+---+---+---+
|               |
+   +   +---+---+
|       |       |
+---+---+   +   +
|               |
+---+---+---+---+
+---+---+---+---+
|               |
+   +---+   +---+
|   |   |   |   |
+---+   +---+   +
|               |
+---+---+---+---+
+---+---+---+---+
|               |
+---+   +---+   +
|   |   |   |   |
+   +---+   +---+
|               |
+---+---+---+---+
+---+---+---+---+
|               |
+---+---+   +   +
|       |       |
+   +   +---+---+
|               |
+---+---+---+---+
+---+---+---+---+
|           |   |
+   +   +---+   +
|       |       |
+   +---+   +   +
|   |           |
+---+---+---+---+
+---+---+---+---+
|           |   |
+   +---+   +   +
|   |   |   |   |
+   +   +---+   +
|   |           |
+---+---+---+---+
+---+---+---+---+
|       |       |
+   +   +   +   +
|       |       |
+   +   +   +   +
|       |       |
+---+---+---+---+
+---+---+---+---+
|   |           |
+   +---+   +   +
|       |       |
+   +   +---+   +
|           |   |
+---+---+---+---+
+---+---+---+---+
|   |           |
+   +   +---+   +
|   |   |   |   |
+   +---+   +   +
|           |   |
+---+---+---+---+
9

Go[edit]

Translation of: C
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))
}
}
}
}
Output:
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[edit]

Translation of: 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.

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[edit]

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^:[email protected]

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:

   '.#' <"2@:{~ all 3 4
┌────┬────┬────┬────┬────┬────┬────┬────┬────┐
│.###│.###│..##│...#│...#│....│....│....│....│
│.#.#│..##│..##│..##│.#.#│..##│.#.#│#.#.│##..│
│...#│...#│..##│.###│.###│####│####│####│####│
└────┴────┴────┴────┴────┴────┴────┴────┴────┘
$ all 4 5
39 4 5
3 13$ '.#' <"2@:{~ all 4 5
┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
│.####│.####│.####│.####│.####│.####│..###│..###│..###│..###│..###│...##│...##│
│.####│.##.#│.#..#│..###│...##│....#│.####│.##.#│..###│...##│....#│.####│..###│
│....#│.#..#│.##.#│...##│..###│.####│....#│.#..#│...##│..###│.####│....#│...##│
│....#│....#│....#│....#│....#│....#│...##│...##│...##│...##│...##│..###│..###│
├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤
│...##│...##│...##│....#│....#│....#│....#│....#│....#│.....│.....│.....│.....│
│...##│....#│.#..#│.####│..###│...##│....#│.#..#│.##.#│.####│..###│...##│....#│
│..###│.####│.##.#│....#│...##│..###│.####│.##.#│.#..#│....#│...##│..###│.####│
│..###│..###│..###│.####│.####│.####│.####│.####│.####│#####│#####│#####│#####│
├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤
│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│
│.#..#│.##.#│..##.│...#.│.....│.#...│.##..│#.##.│#..#.│#....│##...│###..│####.│
│.##.#│.#..#│#..##│#.###│#####│###.#│##..#│#..#.│#.##.│####.│###..│##...│#....│
│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│
└─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘

Java[edit]

Works with: Java version 7
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<Integer> 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();
}
}
[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]

Kotlin[edit]

Translation of: C
// 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]++
for (i in 0..3)
if (grid[t + next[i]] == 0.toByte())
walk(y + dir[i][0], x + dir[i][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<String>) {
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)}")
}
Output:
 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[edit]

Translation of: C

Output is identical to C's.

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[edit]

Translation of: 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.

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 <= 10; $y++) {
loop ($x = 1; $x <= $y; $x++) {
if (!($x +& 1) || !($y +& 1)) {
printf("%d x %d: %d\n", $y, $x, solve($y, $x, 1));
}
}
}
Output:
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

Python[edit]

Translation of: D
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:

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[edit]

Translation of: D
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[edit]

 
#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)
 
Output:
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[edit]

idiomatic[edit]

/*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
output   when using the default input:
        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[edit]

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 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 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
output   is the same as the idiomatic version.


Ruby[edit]

Translation of: Python
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
Output:
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[edit]

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
Output:
+---+---+
|   |   |
+   +   +
|   |   |
+---+---+
+---+---+
|       |
+---+---+
|       |
+---+---+
2
+---+---+---+---+
|           |   |
+   +   +---+   +
|       |       |
+   +---+   +   +
|   |           |
+---+---+---+---+
+---+---+---+---+
|       |       |
+   +   +   +   +
|       |       |
+   +   +   +   +
|       |       |
+---+---+---+---+
+---+---+---+---+
|           |   |
+   +---+   +   +
|   |   |   |   |
+   +   +---+   +
|   |           |
+---+---+---+---+
+---+---+---+---+
|               |
+   +   +---+---+
|       |       |
+---+---+   +   +
|               |
+---+---+---+---+
+---+---+---+---+
|               |
+   +---+   +---+
|   |   |   |   |
+---+   +---+   +
|               |
+---+---+---+---+
+---+---+---+---+
|   |           |
+   +---+   +   +
|       |       |
+   +   +---+   +
|           |   |
+---+---+---+---+
+---+---+---+---+
|   |           |
+   +   +---+   +
|   |   |   |   |
+   +---+   +   +
|           |   |
+---+---+---+---+
+---+---+---+---+
|               |
+---+   +---+   +
|   |   |   |   |
+   +---+   +---+
|               |
+---+---+---+---+
+---+---+---+---+
|               |
+---+---+   +   +
|       |       |
+   +   +---+---+
|               |
+---+---+---+---+
9

Rust[edit]

Translation of: Python
 
fn cwalk(mut vis: &mut Vec<Vec<bool>>, 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[edit]

Translation of: C
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[edit]

Translation of: Ruby
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).

foreach w,h in ([1..9],[1..w]){
if((w*h).isEven) println("%d x %d: %d".fmt(w, h, cut_it(w,h)));
}
Output:

Output is identical.

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