Cut a rectangle
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.
You are encouraged to solve this task according to the task description, using any language you may know.
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. <lang c>#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; }</lang>output<lang>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</lang>
More awkward solution: after compiling, run ./a.out -v [width] [height] for display of cuts.
<lang 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; }</lang>
Common Lisp
Count only. <lang lisp>(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)))))</lang>output<lang>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</lang>
D
<lang 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));
}</lang>
- 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
<lang 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 </lang> <lang Eiffel> 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 </lang> <lang Eiffel> 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 </lang>
- 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
Count only
<lang elixir>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)</lang>
- 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
<lang elixir>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</lang>
- Output:
+---+---+ | | +---+---+ | | +---+---+ +---+---+ | | | + + + | | | +---+---+ 2 +---+---+---+---+ | | + + +---+---+ | | | +---+---+ + + | | +---+---+---+---+ +---+---+---+---+ | | + +---+ +---+ | | | | | +---+ +---+ + | | +---+---+---+---+ +---+---+---+---+ | | +---+ +---+ + | | | | | + +---+ +---+ | | +---+---+---+---+ +---+---+---+---+ | | +---+---+ + + | | | + + +---+---+ | | +---+---+---+---+ +---+---+---+---+ | | | + + +---+ + | | | + +---+ + + | | | +---+---+---+---+ +---+---+---+---+ | | | + +---+ + + | | | | | + + +---+ + | | | +---+---+---+---+ +---+---+---+---+ | | | + + + + + | | | + + + + + | | | +---+---+---+---+ +---+---+---+---+ | | | + +---+ + + | | | + + +---+ + | | | +---+---+---+---+ +---+---+---+---+ | | | + + +---+ + | | | | | + +---+ + + | | | +---+---+---+---+ 9
Go
<lang 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))
}
}
}
}</lang>
- 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
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. <lang 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] ]
</lang> With GHC -O3 the run-time is about 39 times the D entry.
J
<lang 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</lang>
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:
<lang j> '.#' <"2@:{~ all 3 4 ┌────┬────┬────┬────┬────┬────┬────┬────┬────┐ │.###│.###│..##│...#│...#│....│....│....│....│ │.#.#│..##│..##│..##│.#.#│..##│.#.#│#.#.│##..│ │...#│...#│..##│.###│.###│####│####│####│####│ └────┴────┴────┴────┴────┴────┴────┴────┴────┘
$ all 4 5
39 4 5
3 13$ '.#' <"2@:{~ all 4 5
┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐ │.####│.####│.####│.####│.####│.####│..###│..###│..###│..###│..###│...##│...##│ │.####│.##.#│.#..#│..###│...##│....#│.####│.##.#│..###│...##│....#│.####│..###│ │....#│.#..#│.##.#│...##│..###│.####│....#│.#..#│...##│..###│.####│....#│...##│ │....#│....#│....#│....#│....#│....#│...##│...##│...##│...##│...##│..###│..###│ ├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ │...##│...##│...##│....#│....#│....#│....#│....#│....#│.....│.....│.....│.....│ │...##│....#│.#..#│.####│..###│...##│....#│.#..#│.##.#│.####│..###│...##│....#│ │..###│.####│.##.#│....#│...##│..###│.####│.##.#│.#..#│....#│...##│..###│.####│ │..###│..###│..###│.####│.####│.####│.####│.####│.####│#####│#####│#####│#####│ ├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤ │.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│.....│ │.#..#│.##.#│..##.│...#.│.....│.#...│.##..│#.##.│#..#.│#....│##...│###..│####.│ │.##.#│.#..#│#..##│#.###│#####│###.#│##..#│#..#.│#.##.│####.│###..│##...│#....│ │#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│#####│ └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘</lang>
Java
<lang 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<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();
}
}</lang>
[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
<lang 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]++
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)}: ${RectangleCutter.solve(y, x, true)}")
}</lang>
- Output:
2 x 1: 1 2 x 2: 2 3 x 2: 3 4 x 1: 1 4 x 2: 4 4 x 3: 9 4 x 4: 22 5 x 2: 5 5 x 4: 39 6 x 1: 1 6 x 2: 6 6 x 3: 23 6 x 4: 90 6 x 5: 263 6 x 6: 1018 7 x 2: 7 7 x 4: 151 7 x 6: 2947 8 x 1: 1 8 x 2: 8 8 x 3: 53 8 x 4: 340 8 x 5: 1675 8 x 6: 11174 8 x 7: 55939 8 x 8: 369050 9 x 2: 9 9 x 4: 553 9 x 6: 31721 9 x 8: 1812667 10 x 1: 1 10 x 2: 10 10 x 3: 115 10 x 4: 1228 10 x 5: 10295 10 x 6: 118276 10 x 7: 1026005 10 x 8: 11736888 10 x 9: 99953769 10 x 10: 1124140214
Perl
Output is identical to C's. <lang perl>use strict; use warnings; my @grid = 0;
my ($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, $x, 1)); } }
}
}
MAIN();</lang>
Perl 6
This is a very dumb, straightforward translation of the C code. It is very slow so we'll interrupt the execution and display the partial output.
<lang perl6>subset Byte of Int where ^256; my @grid of Byte = 0;
my Int ($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 <= 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));
}
}
}</lang>
- 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
<lang 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()</lang> 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
<lang 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()</lang> The output is the same.
Racket
<lang 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) </lang>
- 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
idiomatic
<lang 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</lang>
- 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
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).
<lang 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 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</lang>
- output is the same as the idiomatic version.
Ruby
<lang 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
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|
nxty+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</lang>
- Output:
+---+---+ | | | + + + | | | +---+---+ +---+---+ | | +---+---+ | | +---+---+ 2 +---+---+---+---+ | | | + + +---+ + | | | + +---+ + + | | | +---+---+---+---+ +---+---+---+---+ | | | + + + + + | | | + + + + + | | | +---+---+---+---+ +---+---+---+---+ | | | + +---+ + + | | | | | + + +---+ + | | | +---+---+---+---+ +---+---+---+---+ | | + + +---+---+ | | | +---+---+ + + | | +---+---+---+---+ +---+---+---+---+ | | + +---+ +---+ | | | | | +---+ +---+ + | | +---+---+---+---+ +---+---+---+---+ | | | + +---+ + + | | | + + +---+ + | | | +---+---+---+---+ +---+---+---+---+ | | | + + +---+ + | | | | | + +---+ + + | | | +---+---+---+---+ +---+---+---+---+ | | +---+ +---+ + | | | | | + +---+ +---+ | | +---+---+---+---+ +---+---+---+---+ | | +---+---+ + + | | | + + +---+---+ | | +---+---+---+---+ 9
Rust
<lang rust> 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));
}
}
}
} </lang>
Tcl
<lang 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</lang> Output is identical.
zkl
<lang 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);
}
}</lang> 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). <lang 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)));
}</lang>
- 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