# Project Euler 086
# Least M such that solutions for shortest cuboid routes ≤ M exceed one million.
function is_square(n: i64) -> bool {
if n <= 0 { return false }
let mut r: i64 = 1
# integer sqrt via Newton
let mut x: i64 = n
while x > 1 {
let y: i64 = (x + n / x) / 2
if y >= x { break }
x = y
}
r = x
while r * r > n {
r = r - 1
}
while (r + 1) * (r + 1) <= n {
r = r + 1
}
return r * r == n
}
function count_for_m(m: i64) -> i64 {
# for cuboid a≤b≤c=m, shortest is min path on faces:
# sqrt((a+b)^2 + c^2) is shortest when a,b ≤ c (the usual case for a≤b≤c)
# number of a,b with 1≤a≤b≤m and (a+b)^2+m^2 square
let mut cnt: i64 = 0
let mut s: i64 = 2
while s <= 2 * m {
if is_square(s * s + m * m) {
# a+b = s, 1≤a≤b≤m, a+b=s ⇒ max(1,s-m) ≤ a ≤ min(s/2, s-1) and b=s-a≤m
let mut lo: i64 = s - m
if lo < 1 { lo = 1 }
let mut hi: i64 = s / 2
if hi > m { hi = m }
# also b = s-a ≥ a and b ≤ m already in bounds
if hi >= lo {
cnt = cnt + (hi - lo + 1)
}
}
s = s + 1
}
return cnt
}
function main() -> i32 {
let mut total: i64 = 0
let mut m: i64 = 0
while total <= 1000000 {
m = m + 1
total = total + count_for_m(m)
}
printf("%lld\n", m)
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; }
bool is_square_i64(int64_t n);
int64_t count_for_m_i64(int64_t m);
int32_t main(void);
bool is_square_i64(int64_t n) {
if (n <= 0) {
return 0;
}
int64_t r = 1;
int64_t x = n;
while (x > 1) {
int64_t y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
if (y >= x) {
break;
}
x = y;
}
r = x;
while ((r * r) > n) {
r = (r - 1);
}
while (((r + 1) * (r + 1)) <= n) {
r = (r + 1);
}
return (r * r) == n;
}
int64_t count_for_m_i64(int64_t m) {
int64_t cnt = 0;
int64_t s = 2;
while (s <= (2 * m)) {
if (is_square_i64(((s * s) + (m * m)))) {
int64_t lo = (s - m);
if (lo < 1) {
lo = 1;
}
int64_t hi = FLOW_CHECKED_DIV((s), (2));
if (hi > m) {
hi = m;
}
if (hi >= lo) {
cnt = (cnt + ((hi - lo) + 1));
}
}
s = (s + 1);
}
return cnt;
}
int32_t main(void) {
int64_t total = 0;
int64_t m = 0;
while (total <= 1000000) {
m = (m + 1);
total = (total + count_for_m_i64(m));
}
printf("%lld\n", m);
return 0;
}