Shut the Box: expected turns for Bob to flip all 12 cards face down under optimal play. State = bitmask of face-up cards. Each roll (x, y) lets Bob toggle card x, y, or x+y; the Bellman equation V(s) = 1 + E[min choice V(s')] is a stochastic shortest path problem, solved by value iteration from V = 0 (monotone convergence; the check game with two coins and four cards gives 5.673651 as stated).
# Project Euler 640
# Shut the Box: expected turns for Bob to flip all 12 cards face down
# under optimal play.
#
# State = bitmask of face-up cards. Each roll (x, y) lets Bob toggle card
# x, y, or x+y; the Bellman equation V(s) = 1 + E[min choice V(s')] is a
# stochastic shortest path problem, solved by value iteration from V = 0
# (monotone convergence; the check game with two coins and four cards
# gives 5.673651 as stated).
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function main() -> i32 {
let N: i64 = 4096
let V: ptr<f64> = calloc(N, 8)
let Vn: ptr<f64> = calloc(N, 8)
let mut it: i64 = 0
let mut diff: f64 = 1.0
while it < 20000 && diff > 0.0000000000001 {
diff = 0.0
let mut s: i64 = 1
while s < N {
let mut tot: f64 = 0.0
let mut x: i64 = 1
while x <= 6 {
let mut y: i64 = 1
while y <= 6 {
# choices: toggle card x, y, or x+y (cards 1..12)
let mut best: f64 = V[s ^ (1 << (x - 1))]
let vy: f64 = V[s ^ (1 << (y - 1))]
if vy < best {
best = vy
}
let vxy: f64 = V[s ^ (1 << (x + y - 1))]
if vxy < best {
best = vxy
}
tot = tot + best
y = y + 1
}
x = x + 1
}
Vn[s] = 1.0 + tot / 36.0
s = s + 1
}
s = 1
while s < N {
let d: f64 = Vn[s] - V[s]
let mut ad: f64 = d
if ad < 0.0 {
ad = 0.0 - ad
}
if ad > diff {
diff = ad
}
V[s] = Vn[s]
s = s + 1
}
it = it + 1
}
printf("%.6f\n", V[N - 1])
free(V)
free(Vn)
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int32_t main(void);
int32_t main(void) {
int64_t N = 4096;
double* V = (double*)(calloc(N, 8));
double* Vn = (double*)(calloc(N, 8));
int64_t it = 0;
double diff = 1.0;
while ((it < 20000 && diff > 0.0000000000001)) {
diff = 0.0;
int64_t s = 1;
while (s < N) {
double tot = 0.0;
int64_t x = 1;
while (x <= 6) {
int64_t y = 1;
while (y <= 6) {
double best = V[(s ^ FLOW_CHECKED_SHL((1), ((x - 1))))];
double vy = V[(s ^ FLOW_CHECKED_SHL((1), ((y - 1))))];
if (vy < best) {
best = vy;
}
double vxy = V[(s ^ FLOW_CHECKED_SHL((1), (((x + y) - 1))))];
if (vxy < best) {
best = vxy;
}
tot = (tot + best);
y = (y + 1);
}
x = (x + 1);
}
Vn[s] = (1.0 + (tot / 36.0));
s = (s + 1);
}
s = 1;
while (s < N) {
double d = (Vn[s] - V[s]);
double ad = d;
if (ad < 0.0) {
ad = (0.0 - ad);
}
if (ad > diff) {
diff = ad;
}
V[s] = Vn[s];
s = (s + 1);
}
it = (it + 1);
}
printf("%.6f\n", V[(N - 1)]);
free(V);
free(Vn);
return 0;
}