f278dcf4 |
1 | /* |
2 | * This program implements a breadth-first search which |
3 | * exhaustively solves the Countdown numbers game, and related |
4 | * games with slightly different rule sets such as `Flippo'. |
5 | * |
6 | * Currently it is simply a standalone command-line utility to |
7 | * which you provide a set of numbers and it tells you everything |
8 | * it can make together with how many different ways it can be |
9 | * made. I would like ultimately to turn it into the generator for |
10 | * a Puzzles puzzle, but I haven't even started on writing a |
11 | * Puzzles user interface yet. |
12 | */ |
13 | |
14 | /* |
15 | * TODO: |
16 | * |
17 | * - start thinking about difficulty ratings |
18 | * + anything involving associative operations will be flagged |
19 | * as many-paths because of the associative options (e.g. |
20 | * 2*3*4 can be (2*3)*4 or 2*(3*4), or indeed (2*4)*3). This |
21 | * is probably a _good_ thing, since those are unusually |
22 | * easy. |
23 | * + tree-structured calculations ((a*b)/(c+d)) have multiple |
24 | * paths because the independent branches of the tree can be |
25 | * evaluated in either order, whereas straight-line |
26 | * calculations with no branches will be considered easier. |
27 | * Can we do anything about this? It's certainly not clear to |
28 | * me that tree-structure calculations are _easier_, although |
29 | * I'm also not convinced they're harder. |
30 | * + I think for a realistic difficulty assessment we must also |
31 | * consider the `obviousness' of the arithmetic operations in |
32 | * some heuristic sense, and also (in Countdown) how many |
33 | * numbers ended up being used. |
34 | * - actually try some generations |
35 | * - at this point we're probably ready to start on the Puzzles |
36 | * integration. |
37 | */ |
38 | |
39 | #include <stdio.h> |
ed35622c |
40 | #include <string.h> |
f278dcf4 |
41 | #include <limits.h> |
42 | #include <assert.h> |
ed35622c |
43 | #include <math.h> |
f278dcf4 |
44 | |
45 | #include "puzzles.h" |
46 | #include "tree234.h" |
47 | |
48 | /* |
49 | * To search for numbers we can make, we employ a breadth-first |
50 | * search across the space of sets of input numbers. That is, for |
51 | * example, we start with the set (3,6,25,50,75,100); we apply |
52 | * moves which involve combining two numbers (e.g. adding the 50 |
53 | * and the 75 takes us to the set (3,6,25,100,125); and then we see |
54 | * if we ever end up with a set containing (say) 952. |
55 | * |
56 | * If the rules are changed so that all the numbers must be used, |
57 | * this is easy to adjust to: we simply see if we end up with a set |
58 | * containing _only_ (say) 952. |
59 | * |
60 | * Obviously, we can vary the rules about permitted arithmetic |
61 | * operations simply by altering the set of valid moves in the bfs. |
62 | * However, there's one common rule in this sort of puzzle which |
63 | * takes a little more thought, and that's _concatenation_. For |
64 | * example, if you are given (say) four 4s and required to make 10, |
65 | * you are permitted to combine two of the 4s into a 44 to begin |
66 | * with, making (44-4)/4 = 10. However, you are generally not |
67 | * allowed to concatenate two numbers that _weren't_ both in the |
68 | * original input set (you couldn't multiply two 4s to get 16 and |
69 | * then concatenate a 4 on to it to make 164), so concatenation is |
70 | * not an operation which is valid in all situations. |
71 | * |
72 | * We could enforce this restriction by storing a flag alongside |
73 | * each number indicating whether or not it's an original number; |
74 | * the rules being that concatenation of two numbers is only valid |
75 | * if they both have the original flag, and that its output _also_ |
76 | * has the original flag (so that you can concatenate three 4s into |
77 | * a 444), but that applying any other arithmetic operation clears |
78 | * the original flag on the output. However, we can get marginally |
79 | * simpler than that by observing that since concatenation has to |
80 | * happen to a number before any other operation, we can simply |
81 | * place all the concatenations at the start of the search. In |
82 | * other words, we have a global flag on an entire number _set_ |
83 | * which indicates whether we are still permitted to perform |
84 | * concatenations; if so, we can concatenate any of the numbers in |
85 | * that set. Performing any other operation clears the flag. |
86 | */ |
87 | |
88 | #define SETFLAG_CONCAT 1 /* we can do concatenation */ |
89 | |
90 | struct sets; |
91 | |
ed35622c |
92 | struct ancestor { |
93 | struct set *prev; /* index of ancestor set in set list */ |
94 | unsigned char pa, pb, po, pr; /* operation that got here from prev */ |
95 | }; |
96 | |
f278dcf4 |
97 | struct set { |
98 | int *numbers; /* rationals stored as n,d pairs */ |
99 | short nnumbers; /* # of rationals, so half # of ints */ |
100 | short flags; /* SETFLAG_CONCAT only, at present */ |
f278dcf4 |
101 | int npaths; /* number of ways to reach this set */ |
ed35622c |
102 | struct ancestor a; /* primary ancestor */ |
103 | struct ancestor *as; /* further ancestors, if we care */ |
104 | int nas, assize; |
f278dcf4 |
105 | }; |
106 | |
107 | struct output { |
108 | int number; |
109 | struct set *set; |
110 | int index; /* which number in the set is it? */ |
111 | int npaths; /* number of ways to reach this */ |
112 | }; |
113 | |
114 | #define SETLISTLEN 1024 |
115 | #define NUMBERLISTLEN 32768 |
116 | #define OUTPUTLISTLEN 1024 |
117 | struct operation; |
118 | struct sets { |
119 | struct set **setlists; |
120 | int nsets, nsetlists, setlistsize; |
121 | tree234 *settree; |
122 | int **numberlists; |
123 | int nnumbers, nnumberlists, numberlistsize; |
124 | struct output **outputlists; |
125 | int noutputs, noutputlists, outputlistsize; |
126 | tree234 *outputtree; |
127 | const struct operation *const *ops; |
128 | }; |
129 | |
130 | #define OPFLAG_NEEDS_CONCAT 1 |
131 | #define OPFLAG_KEEPS_CONCAT 2 |
ed35622c |
132 | #define OPFLAG_UNARY 4 |
688f45ad |
133 | #define OPFLAG_UNARYPREFIX 8 |
f278dcf4 |
134 | |
135 | struct operation { |
136 | /* |
137 | * Most operations should be shown in the output working, but |
138 | * concatenation should not; we just take the result of the |
139 | * concatenation and assume that it's obvious how it was |
140 | * derived. |
141 | */ |
142 | int display; |
143 | |
144 | /* |
91623748 |
145 | * Text display of the operator, in expressions and for |
146 | * debugging respectively. |
f278dcf4 |
147 | */ |
91623748 |
148 | char *text, *dbgtext; |
f278dcf4 |
149 | |
150 | /* |
151 | * Flags dictating when the operator can be applied. |
152 | */ |
153 | int flags; |
154 | |
155 | /* |
156 | * Priority of the operator (for avoiding unnecessary |
157 | * parentheses when formatting it into a string). |
158 | */ |
159 | int priority; |
160 | |
161 | /* |
162 | * Associativity of the operator. Bit 0 means we need parens |
163 | * when the left operand of one of these operators is another |
164 | * instance of it, e.g. (2^3)^4. Bit 1 means we need parens |
165 | * when the right operand is another instance of the same |
166 | * operator, e.g. 2-(3-4). Thus: |
167 | * |
168 | * - this field is 0 for a fully associative operator, since |
169 | * we never need parens. |
170 | * - it's 1 for a right-associative operator. |
171 | * - it's 2 for a left-associative operator. |
172 | * - it's 3 for a _non_-associative operator (which always |
173 | * uses parens just to be sure). |
174 | */ |
175 | int assoc; |
176 | |
177 | /* |
178 | * Whether the operator is commutative. Saves time in the |
179 | * search if we don't have to try it both ways round. |
180 | */ |
181 | int commutes; |
182 | |
183 | /* |
184 | * Function which implements the operator. Returns TRUE on |
185 | * success, FALSE on failure. Takes two rationals and writes |
186 | * out a third. |
187 | */ |
188 | int (*perform)(int *a, int *b, int *output); |
189 | }; |
190 | |
191 | struct rules { |
192 | const struct operation *const *ops; |
193 | int use_all; |
194 | }; |
195 | |
196 | #define MUL(r, a, b) do { \ |
197 | (r) = (a) * (b); \ |
198 | if ((b) && (a) && (r) / (b) != (a)) return FALSE; \ |
199 | } while (0) |
200 | |
201 | #define ADD(r, a, b) do { \ |
202 | (r) = (a) + (b); \ |
203 | if ((a) > 0 && (b) > 0 && (r) < 0) return FALSE; \ |
204 | if ((a) < 0 && (b) < 0 && (r) > 0) return FALSE; \ |
205 | } while (0) |
206 | |
207 | #define OUT(output, n, d) do { \ |
208 | int g = gcd((n),(d)); \ |
ed35622c |
209 | if (g < 0) g = -g; \ |
f278dcf4 |
210 | if ((d) < 0) g = -g; \ |
ed35622c |
211 | if (g == -1 && (n) < -INT_MAX) return FALSE; \ |
212 | if (g == -1 && (d) < -INT_MAX) return FALSE; \ |
f278dcf4 |
213 | (output)[0] = (n)/g; \ |
214 | (output)[1] = (d)/g; \ |
215 | assert((output)[1] > 0); \ |
216 | } while (0) |
217 | |
218 | static int gcd(int x, int y) |
219 | { |
220 | while (x != 0 && y != 0) { |
221 | int t = x; |
222 | x = y; |
223 | y = t % y; |
224 | } |
225 | |
226 | return abs(x + y); /* i.e. whichever one isn't zero */ |
227 | } |
228 | |
229 | static int perform_add(int *a, int *b, int *output) |
230 | { |
231 | int at, bt, tn, bn; |
232 | /* |
233 | * a0/a1 + b0/b1 = (a0*b1 + b0*a1) / (a1*b1) |
234 | */ |
235 | MUL(at, a[0], b[1]); |
236 | MUL(bt, b[0], a[1]); |
237 | ADD(tn, at, bt); |
238 | MUL(bn, a[1], b[1]); |
239 | OUT(output, tn, bn); |
240 | return TRUE; |
241 | } |
242 | |
243 | static int perform_sub(int *a, int *b, int *output) |
244 | { |
245 | int at, bt, tn, bn; |
246 | /* |
247 | * a0/a1 - b0/b1 = (a0*b1 - b0*a1) / (a1*b1) |
248 | */ |
249 | MUL(at, a[0], b[1]); |
250 | MUL(bt, b[0], a[1]); |
251 | ADD(tn, at, -bt); |
252 | MUL(bn, a[1], b[1]); |
253 | OUT(output, tn, bn); |
254 | return TRUE; |
255 | } |
256 | |
257 | static int perform_mul(int *a, int *b, int *output) |
258 | { |
259 | int tn, bn; |
260 | /* |
261 | * a0/a1 * b0/b1 = (a0*b0) / (a1*b1) |
262 | */ |
263 | MUL(tn, a[0], b[0]); |
264 | MUL(bn, a[1], b[1]); |
265 | OUT(output, tn, bn); |
266 | return TRUE; |
267 | } |
268 | |
269 | static int perform_div(int *a, int *b, int *output) |
270 | { |
271 | int tn, bn; |
272 | |
273 | /* |
274 | * Division by zero is outlawed. |
275 | */ |
276 | if (b[0] == 0) |
277 | return FALSE; |
278 | |
279 | /* |
280 | * a0/a1 / b0/b1 = (a0*b1) / (a1*b0) |
281 | */ |
282 | MUL(tn, a[0], b[1]); |
283 | MUL(bn, a[1], b[0]); |
284 | OUT(output, tn, bn); |
285 | return TRUE; |
286 | } |
287 | |
288 | static int perform_exact_div(int *a, int *b, int *output) |
289 | { |
290 | int tn, bn; |
291 | |
292 | /* |
293 | * Division by zero is outlawed. |
294 | */ |
295 | if (b[0] == 0) |
296 | return FALSE; |
297 | |
298 | /* |
299 | * a0/a1 / b0/b1 = (a0*b1) / (a1*b0) |
300 | */ |
301 | MUL(tn, a[0], b[1]); |
302 | MUL(bn, a[1], b[0]); |
303 | OUT(output, tn, bn); |
304 | |
305 | /* |
306 | * Exact division means we require the result to be an integer. |
307 | */ |
308 | return (output[1] == 1); |
309 | } |
310 | |
688f45ad |
311 | static int max_p10(int n, int *p10_r) |
312 | { |
313 | /* |
314 | * Find the smallest power of ten strictly greater than n. |
315 | * |
316 | * Special case: we must return at least 10, even if n is |
317 | * zero. (This is because this function is used for finding |
318 | * the power of ten by which to multiply a number being |
319 | * concatenated to the front of n, and concatenating 1 to 0 |
320 | * should yield 10 and not 1.) |
321 | */ |
322 | int p10 = 10; |
323 | while (p10 <= (INT_MAX/10) && p10 <= n) |
324 | p10 *= 10; |
325 | if (p10 > INT_MAX/10) |
326 | return FALSE; /* integer overflow */ |
327 | *p10_r = p10; |
328 | return TRUE; |
329 | } |
330 | |
f278dcf4 |
331 | static int perform_concat(int *a, int *b, int *output) |
332 | { |
333 | int t1, t2, p10; |
334 | |
335 | /* |
ed35622c |
336 | * We can't concatenate anything which isn't a non-negative |
337 | * integer. |
f278dcf4 |
338 | */ |
ed35622c |
339 | if (a[1] != 1 || b[1] != 1 || a[0] < 0 || b[0] < 0) |
f278dcf4 |
340 | return FALSE; |
341 | |
342 | /* |
343 | * For concatenation, we can safely assume leading zeroes |
344 | * aren't an issue. It isn't clear whether they `should' be |
345 | * allowed, but it turns out not to matter: concatenating a |
346 | * leading zero on to a number in order to harmlessly get rid |
347 | * of the zero is never necessary because unwanted zeroes can |
348 | * be disposed of by adding them to something instead. So we |
349 | * disallow them always. |
350 | * |
351 | * The only other possibility is that you might want to |
352 | * concatenate a leading zero on to something and then |
353 | * concatenate another non-zero digit on to _that_ (to make, |
354 | * for example, 106); but that's also unnecessary, because you |
355 | * can make 106 just as easily by concatenating the 0 on to the |
356 | * _end_ of the 1 first. |
357 | */ |
358 | if (a[0] == 0) |
359 | return FALSE; |
360 | |
688f45ad |
361 | if (!max_p10(b[0], &p10)) return FALSE; |
362 | |
f278dcf4 |
363 | MUL(t1, p10, a[0]); |
364 | ADD(t2, t1, b[0]); |
365 | OUT(output, t2, 1); |
366 | return TRUE; |
367 | } |
368 | |
ed35622c |
369 | #define IPOW(ret, x, y) do { \ |
370 | int ipow_limit = (y); \ |
371 | if ((x) == 1 || (x) == 0) ipow_limit = 1; \ |
372 | else if ((x) == -1) ipow_limit &= 1; \ |
373 | (ret) = 1; \ |
374 | while (ipow_limit-- > 0) { \ |
375 | int tmp; \ |
376 | MUL(tmp, ret, x); \ |
377 | ret = tmp; \ |
378 | } \ |
379 | } while (0) |
380 | |
381 | static int perform_exp(int *a, int *b, int *output) |
382 | { |
688f45ad |
383 | int an, ad, xn, xd; |
ed35622c |
384 | |
385 | /* |
386 | * Exponentiation is permitted if the result is rational. This |
387 | * means that: |
388 | * |
389 | * - first we see whether we can take the (denominator-of-b)th |
390 | * root of a and get a rational; if not, we give up. |
391 | * |
392 | * - then we do take that root of a |
393 | * |
394 | * - then we multiply by itself (numerator-of-b) times. |
395 | */ |
396 | if (b[1] > 1) { |
688f45ad |
397 | an = (int)(0.5 + pow(a[0], 1.0/b[1])); |
398 | ad = (int)(0.5 + pow(a[1], 1.0/b[1])); |
ed35622c |
399 | IPOW(xn, an, b[1]); |
400 | IPOW(xd, ad, b[1]); |
401 | if (xn != a[0] || xd != a[1]) |
402 | return FALSE; |
403 | } else { |
404 | an = a[0]; |
405 | ad = a[1]; |
406 | } |
407 | if (b[0] >= 0) { |
408 | IPOW(xn, an, b[0]); |
409 | IPOW(xd, ad, b[0]); |
410 | } else { |
411 | IPOW(xd, an, -b[0]); |
412 | IPOW(xn, ad, -b[0]); |
413 | } |
414 | if (xd == 0) |
415 | return FALSE; |
416 | |
417 | OUT(output, xn, xd); |
418 | return TRUE; |
419 | } |
420 | |
421 | static int perform_factorial(int *a, int *b, int *output) |
422 | { |
423 | int ret, t, i; |
424 | |
425 | /* |
426 | * Factorials of non-negative integers are permitted. |
427 | */ |
428 | if (a[1] != 1 || a[0] < 0) |
429 | return FALSE; |
430 | |
91623748 |
431 | /* |
432 | * However, a special case: we don't take a factorial of |
433 | * anything which would thereby remain the same. |
434 | */ |
435 | if (a[0] == 1 || a[0] == 2) |
436 | return FALSE; |
437 | |
ed35622c |
438 | ret = 1; |
439 | for (i = 1; i <= a[0]; i++) { |
440 | MUL(t, ret, i); |
441 | ret = t; |
442 | } |
443 | |
444 | OUT(output, ret, 1); |
445 | return TRUE; |
446 | } |
447 | |
688f45ad |
448 | static int perform_decimal(int *a, int *b, int *output) |
449 | { |
450 | int p10; |
451 | |
452 | /* |
453 | * Add a decimal digit to the front of a number; |
454 | * fail if it's not an integer. |
455 | * So, 1 --> 0.1, 15 --> 0.15, |
456 | * or, rather, 1 --> 1/10, 15 --> 15/100, |
457 | * x --> x / (smallest power of 10 > than x) |
458 | * |
459 | */ |
460 | if (a[1] != 1) return FALSE; |
461 | |
462 | if (!max_p10(a[0], &p10)) return FALSE; |
463 | |
464 | OUT(output, a[0], p10); |
465 | return TRUE; |
466 | } |
467 | |
468 | static int perform_recur(int *a, int *b, int *output) |
469 | { |
470 | int p10, tn, bn; |
471 | |
472 | /* |
473 | * This converts a number like .4 to .44444..., or .45 to .45454... |
474 | * The input number must be -1 < a < 1. |
475 | * |
476 | * Calculate the smallest power of 10 that divides the denominator exactly, |
477 | * returning if no such power of 10 exists. Then multiply the numerator |
478 | * up accordingly, and the new denominator becomes that power of 10 - 1. |
479 | */ |
480 | if (abs(a[0]) >= abs(a[1])) return FALSE; /* -1 < a < 1 */ |
481 | |
482 | p10 = 10; |
483 | while (p10 <= (INT_MAX/10)) { |
484 | if ((a[1] <= p10) && (p10 % a[1]) == 0) goto found; |
485 | p10 *= 10; |
486 | } |
487 | return FALSE; |
488 | found: |
489 | tn = a[0] * (p10 / a[1]); |
490 | bn = p10 - 1; |
491 | |
492 | OUT(output, tn, bn); |
493 | return TRUE; |
494 | } |
495 | |
496 | static int perform_root(int *a, int *b, int *output) |
497 | { |
498 | /* |
499 | * A root B is: 1 iff a == 0 |
500 | * B ^ (1/A) otherwise |
501 | */ |
502 | int ainv[2], res; |
503 | |
504 | if (a[0] == 0) { |
505 | OUT(output, 1, 1); |
506 | return TRUE; |
507 | } |
508 | |
509 | OUT(ainv, a[1], a[0]); |
510 | res = perform_exp(b, ainv, output); |
511 | return res; |
512 | } |
513 | |
f278dcf4 |
514 | const static struct operation op_add = { |
91623748 |
515 | TRUE, "+", "+", 0, 10, 0, TRUE, perform_add |
f278dcf4 |
516 | }; |
517 | const static struct operation op_sub = { |
91623748 |
518 | TRUE, "-", "-", 0, 10, 2, FALSE, perform_sub |
f278dcf4 |
519 | }; |
520 | const static struct operation op_mul = { |
91623748 |
521 | TRUE, "*", "*", 0, 20, 0, TRUE, perform_mul |
f278dcf4 |
522 | }; |
523 | const static struct operation op_div = { |
91623748 |
524 | TRUE, "/", "/", 0, 20, 2, FALSE, perform_div |
f278dcf4 |
525 | }; |
526 | const static struct operation op_xdiv = { |
91623748 |
527 | TRUE, "/", "/", 0, 20, 2, FALSE, perform_exact_div |
f278dcf4 |
528 | }; |
529 | const static struct operation op_concat = { |
91623748 |
530 | FALSE, "", "concat", OPFLAG_NEEDS_CONCAT | OPFLAG_KEEPS_CONCAT, |
f278dcf4 |
531 | 1000, 0, FALSE, perform_concat |
532 | }; |
ed35622c |
533 | const static struct operation op_exp = { |
91623748 |
534 | TRUE, "^", "^", 0, 30, 1, FALSE, perform_exp |
ed35622c |
535 | }; |
536 | const static struct operation op_factorial = { |
91623748 |
537 | TRUE, "!", "!", OPFLAG_UNARY, 40, 0, FALSE, perform_factorial |
ed35622c |
538 | }; |
688f45ad |
539 | const static struct operation op_decimal = { |
540 | TRUE, ".", ".", OPFLAG_UNARY | OPFLAG_UNARYPREFIX | OPFLAG_NEEDS_CONCAT | OPFLAG_KEEPS_CONCAT, 50, 0, FALSE, perform_decimal |
541 | }; |
542 | const static struct operation op_recur = { |
543 | TRUE, "...", "recur", OPFLAG_UNARY | OPFLAG_NEEDS_CONCAT, 45, 2, FALSE, perform_recur |
544 | }; |
545 | const static struct operation op_root = { |
546 | TRUE, "v~", "root", 0, 30, 1, FALSE, perform_root |
547 | }; |
f278dcf4 |
548 | |
549 | /* |
550 | * In Countdown, divisions resulting in fractions are disallowed. |
551 | * http://www.askoxford.com/wordgames/countdown/rules/ |
552 | */ |
553 | const static struct operation *const ops_countdown[] = { |
554 | &op_add, &op_mul, &op_sub, &op_xdiv, NULL |
555 | }; |
556 | const static struct rules rules_countdown = { |
557 | ops_countdown, FALSE |
558 | }; |
559 | |
560 | /* |
561 | * A slightly different rule set which handles the reasonably well |
562 | * known puzzle of making 24 using two 3s and two 8s. For this we |
563 | * need rational rather than integer division. |
564 | */ |
565 | const static struct operation *const ops_3388[] = { |
566 | &op_add, &op_mul, &op_sub, &op_div, NULL |
567 | }; |
568 | const static struct rules rules_3388 = { |
569 | ops_3388, TRUE |
570 | }; |
571 | |
572 | /* |
573 | * A still more permissive rule set usable for the four-4s problem |
574 | * and similar things. Permits concatenation. |
575 | */ |
576 | const static struct operation *const ops_four4s[] = { |
577 | &op_add, &op_mul, &op_sub, &op_div, &op_concat, NULL |
578 | }; |
579 | const static struct rules rules_four4s = { |
580 | ops_four4s, TRUE |
581 | }; |
582 | |
ed35622c |
583 | /* |
584 | * The most permissive ruleset I can think of. Permits |
585 | * exponentiation, and also silly unary operators like factorials. |
586 | */ |
587 | const static struct operation *const ops_anythinggoes[] = { |
688f45ad |
588 | &op_add, &op_mul, &op_sub, &op_div, &op_concat, &op_exp, &op_factorial, |
589 | &op_decimal, &op_recur, &op_root, NULL |
ed35622c |
590 | }; |
591 | const static struct rules rules_anythinggoes = { |
592 | ops_anythinggoes, TRUE |
593 | }; |
594 | |
f278dcf4 |
595 | #define ratcmp(a,op,b) ( (long long)(a)[0] * (b)[1] op \ |
596 | (long long)(b)[0] * (a)[1] ) |
597 | |
598 | static int addtoset(struct set *set, int newnumber[2]) |
599 | { |
600 | int i, j; |
601 | |
602 | /* Find where we want to insert the new number */ |
603 | for (i = 0; i < set->nnumbers && |
604 | ratcmp(set->numbers+2*i, <, newnumber); i++); |
605 | |
606 | /* Move everything else up */ |
607 | for (j = set->nnumbers; j > i; j--) { |
608 | set->numbers[2*j] = set->numbers[2*j-2]; |
609 | set->numbers[2*j+1] = set->numbers[2*j-1]; |
610 | } |
611 | |
612 | /* Insert the new number */ |
613 | set->numbers[2*i] = newnumber[0]; |
614 | set->numbers[2*i+1] = newnumber[1]; |
615 | |
616 | set->nnumbers++; |
617 | |
618 | return i; |
619 | } |
620 | |
621 | #define ensure(array, size, newlen, type) do { \ |
622 | if ((newlen) > (size)) { \ |
623 | (size) = (newlen) + 512; \ |
624 | (array) = sresize((array), (size), type); \ |
625 | } \ |
626 | } while (0) |
627 | |
628 | static int setcmp(void *av, void *bv) |
629 | { |
630 | struct set *a = (struct set *)av; |
631 | struct set *b = (struct set *)bv; |
632 | int i; |
633 | |
634 | if (a->nnumbers < b->nnumbers) |
635 | return -1; |
636 | else if (a->nnumbers > b->nnumbers) |
637 | return +1; |
638 | |
639 | if (a->flags < b->flags) |
640 | return -1; |
641 | else if (a->flags > b->flags) |
642 | return +1; |
643 | |
644 | for (i = 0; i < a->nnumbers; i++) { |
645 | if (ratcmp(a->numbers+2*i, <, b->numbers+2*i)) |
646 | return -1; |
647 | else if (ratcmp(a->numbers+2*i, >, b->numbers+2*i)) |
648 | return +1; |
649 | } |
650 | |
651 | return 0; |
652 | } |
653 | |
654 | static int outputcmp(void *av, void *bv) |
655 | { |
656 | struct output *a = (struct output *)av; |
657 | struct output *b = (struct output *)bv; |
658 | |
659 | if (a->number < b->number) |
660 | return -1; |
661 | else if (a->number > b->number) |
662 | return +1; |
663 | |
664 | return 0; |
665 | } |
666 | |
667 | static int outputfindcmp(void *av, void *bv) |
668 | { |
669 | int *a = (int *)av; |
670 | struct output *b = (struct output *)bv; |
671 | |
672 | if (*a < b->number) |
673 | return -1; |
674 | else if (*a > b->number) |
675 | return +1; |
676 | |
677 | return 0; |
678 | } |
679 | |
ed35622c |
680 | static void addset(struct sets *s, struct set *set, int multiple, |
681 | struct set *prev, int pa, int po, int pb, int pr) |
f278dcf4 |
682 | { |
683 | struct set *s2; |
684 | int npaths = (prev ? prev->npaths : 1); |
685 | |
686 | assert(set == s->setlists[s->nsets / SETLISTLEN] + s->nsets % SETLISTLEN); |
687 | s2 = add234(s->settree, set); |
688 | if (s2 == set) { |
689 | /* |
690 | * New set added to the tree. |
691 | */ |
ed35622c |
692 | set->a.prev = prev; |
693 | set->a.pa = pa; |
694 | set->a.po = po; |
695 | set->a.pb = pb; |
696 | set->a.pr = pr; |
f278dcf4 |
697 | set->npaths = npaths; |
698 | s->nsets++; |
699 | s->nnumbers += 2 * set->nnumbers; |
ed35622c |
700 | set->as = NULL; |
701 | set->nas = set->assize = 0; |
f278dcf4 |
702 | } else { |
703 | /* |
ed35622c |
704 | * Rediscovered an existing set. Update its npaths. |
f278dcf4 |
705 | */ |
706 | s2->npaths += npaths; |
ed35622c |
707 | /* |
708 | * And optionally enter it as an additional ancestor. |
709 | */ |
710 | if (multiple) { |
711 | if (s2->nas >= s2->assize) { |
712 | s2->assize = s2->nas * 3 / 2 + 4; |
713 | s2->as = sresize(s2->as, s2->assize, struct ancestor); |
714 | } |
715 | s2->as[s2->nas].prev = prev; |
716 | s2->as[s2->nas].pa = pa; |
717 | s2->as[s2->nas].po = po; |
718 | s2->as[s2->nas].pb = pb; |
719 | s2->as[s2->nas].pr = pr; |
720 | s2->nas++; |
721 | } |
f278dcf4 |
722 | } |
723 | } |
724 | |
725 | static struct set *newset(struct sets *s, int nnumbers, int flags) |
726 | { |
727 | struct set *sn; |
728 | |
729 | ensure(s->setlists, s->setlistsize, s->nsets/SETLISTLEN+1, struct set *); |
730 | while (s->nsetlists <= s->nsets / SETLISTLEN) |
731 | s->setlists[s->nsetlists++] = snewn(SETLISTLEN, struct set); |
732 | sn = s->setlists[s->nsets / SETLISTLEN] + s->nsets % SETLISTLEN; |
733 | |
734 | if (s->nnumbers + nnumbers * 2 > s->nnumberlists * NUMBERLISTLEN) |
735 | s->nnumbers = s->nnumberlists * NUMBERLISTLEN; |
736 | ensure(s->numberlists, s->numberlistsize, |
737 | s->nnumbers/NUMBERLISTLEN+1, int *); |
738 | while (s->nnumberlists <= s->nnumbers / NUMBERLISTLEN) |
739 | s->numberlists[s->nnumberlists++] = snewn(NUMBERLISTLEN, int); |
740 | sn->numbers = s->numberlists[s->nnumbers / NUMBERLISTLEN] + |
741 | s->nnumbers % NUMBERLISTLEN; |
742 | |
743 | /* |
744 | * Start the set off empty. |
745 | */ |
746 | sn->nnumbers = 0; |
747 | |
748 | sn->flags = flags; |
749 | |
750 | return sn; |
751 | } |
752 | |
753 | static int addoutput(struct sets *s, struct set *ss, int index, int *n) |
754 | { |
755 | struct output *o, *o2; |
756 | |
757 | /* |
758 | * Target numbers are always integers. |
759 | */ |
760 | if (ss->numbers[2*index+1] != 1) |
761 | return FALSE; |
762 | |
763 | ensure(s->outputlists, s->outputlistsize, s->noutputs/OUTPUTLISTLEN+1, |
764 | struct output *); |
765 | while (s->noutputlists <= s->noutputs / OUTPUTLISTLEN) |
766 | s->outputlists[s->noutputlists++] = snewn(OUTPUTLISTLEN, |
767 | struct output); |
768 | o = s->outputlists[s->noutputs / OUTPUTLISTLEN] + |
769 | s->noutputs % OUTPUTLISTLEN; |
770 | |
771 | o->number = ss->numbers[2*index]; |
772 | o->set = ss; |
773 | o->index = index; |
774 | o->npaths = ss->npaths; |
775 | o2 = add234(s->outputtree, o); |
776 | if (o2 != o) { |
777 | o2->npaths += o->npaths; |
778 | } else { |
779 | s->noutputs++; |
780 | } |
781 | *n = o->number; |
782 | return TRUE; |
783 | } |
784 | |
785 | static struct sets *do_search(int ninputs, int *inputs, |
ed35622c |
786 | const struct rules *rules, int *target, |
91623748 |
787 | int debug, int multiple) |
f278dcf4 |
788 | { |
789 | struct sets *s; |
790 | struct set *sn; |
791 | int qpos, i; |
792 | const struct operation *const *ops = rules->ops; |
793 | |
794 | s = snew(struct sets); |
795 | s->setlists = NULL; |
796 | s->nsets = s->nsetlists = s->setlistsize = 0; |
797 | s->numberlists = NULL; |
798 | s->nnumbers = s->nnumberlists = s->numberlistsize = 0; |
799 | s->outputlists = NULL; |
800 | s->noutputs = s->noutputlists = s->outputlistsize = 0; |
801 | s->settree = newtree234(setcmp); |
802 | s->outputtree = newtree234(outputcmp); |
803 | s->ops = ops; |
804 | |
805 | /* |
806 | * Start with the input set. |
807 | */ |
808 | sn = newset(s, ninputs, SETFLAG_CONCAT); |
809 | for (i = 0; i < ninputs; i++) { |
810 | int newnumber[2]; |
811 | newnumber[0] = inputs[i]; |
812 | newnumber[1] = 1; |
813 | addtoset(sn, newnumber); |
814 | } |
ed35622c |
815 | addset(s, sn, multiple, NULL, 0, 0, 0, 0); |
f278dcf4 |
816 | |
817 | /* |
818 | * Now perform the breadth-first search: keep looping over sets |
819 | * until we run out of steam. |
820 | */ |
821 | qpos = 0; |
822 | while (qpos < s->nsets) { |
823 | struct set *ss = s->setlists[qpos / SETLISTLEN] + qpos % SETLISTLEN; |
824 | struct set *sn; |
825 | int i, j, k, m; |
826 | |
91623748 |
827 | if (debug) { |
828 | int i; |
829 | printf("processing set:"); |
830 | for (i = 0; i < ss->nnumbers; i++) { |
831 | printf(" %d", ss->numbers[2*i]); |
832 | if (ss->numbers[2*i+1] != 1) |
688f45ad |
833 | printf("/%d", ss->numbers[2*i+1]); |
91623748 |
834 | } |
835 | printf("\n"); |
836 | } |
837 | |
f278dcf4 |
838 | /* |
839 | * Record all the valid output numbers in this state. We |
840 | * can always do this if there's only one number in the |
841 | * state; otherwise, we can only do it if we aren't |
842 | * required to use all the numbers in coming to our answer. |
843 | */ |
844 | if (ss->nnumbers == 1 || !rules->use_all) { |
845 | for (i = 0; i < ss->nnumbers; i++) { |
846 | int n; |
847 | |
848 | if (addoutput(s, ss, i, &n) && target && n == *target) |
849 | return s; |
850 | } |
851 | } |
852 | |
853 | /* |
854 | * Try every possible operation from this state. |
855 | */ |
856 | for (k = 0; ops[k] && ops[k]->perform; k++) { |
857 | if ((ops[k]->flags & OPFLAG_NEEDS_CONCAT) && |
858 | !(ss->flags & SETFLAG_CONCAT)) |
859 | continue; /* can't use this operation here */ |
860 | for (i = 0; i < ss->nnumbers; i++) { |
ed35622c |
861 | int jlimit = (ops[k]->flags & OPFLAG_UNARY ? 1 : ss->nnumbers); |
862 | for (j = 0; j < jlimit; j++) { |
f278dcf4 |
863 | int n[2]; |
ed35622c |
864 | int pa, po, pb, pr; |
f278dcf4 |
865 | |
ed35622c |
866 | if (!(ops[k]->flags & OPFLAG_UNARY)) { |
867 | if (i == j) |
868 | continue; /* can't combine a number with itself */ |
869 | if (i > j && ops[k]->commutes) |
870 | continue; /* no need to do this both ways round */ |
871 | } |
f278dcf4 |
872 | if (!ops[k]->perform(ss->numbers+2*i, ss->numbers+2*j, n)) |
873 | continue; /* operation failed */ |
874 | |
875 | sn = newset(s, ss->nnumbers-1, ss->flags); |
876 | |
877 | if (!(ops[k]->flags & OPFLAG_KEEPS_CONCAT)) |
878 | sn->flags &= ~SETFLAG_CONCAT; |
879 | |
880 | for (m = 0; m < ss->nnumbers; m++) { |
ed35622c |
881 | if (m == i || (!(ops[k]->flags & OPFLAG_UNARY) && |
882 | m == j)) |
f278dcf4 |
883 | continue; |
884 | sn->numbers[2*sn->nnumbers] = ss->numbers[2*m]; |
885 | sn->numbers[2*sn->nnumbers + 1] = ss->numbers[2*m + 1]; |
886 | sn->nnumbers++; |
887 | } |
ed35622c |
888 | pa = i; |
889 | if (ops[k]->flags & OPFLAG_UNARY) |
890 | pb = sn->nnumbers+10; |
891 | else |
892 | pb = j; |
893 | po = k; |
894 | pr = addtoset(sn, n); |
895 | addset(s, sn, multiple, ss, pa, po, pb, pr); |
91623748 |
896 | if (debug) { |
897 | int i; |
688f45ad |
898 | if (ops[k]->flags & OPFLAG_UNARYPREFIX) |
899 | printf(" %s %d ->", ops[po]->dbgtext, pa); |
900 | else if (ops[k]->flags & OPFLAG_UNARY) |
901 | printf(" %d %s ->", pa, ops[po]->dbgtext); |
902 | else |
903 | printf(" %d %s %d ->", pa, ops[po]->dbgtext, pb); |
91623748 |
904 | for (i = 0; i < sn->nnumbers; i++) { |
905 | printf(" %d", sn->numbers[2*i]); |
906 | if (sn->numbers[2*i+1] != 1) |
688f45ad |
907 | printf("/%d", sn->numbers[2*i+1]); |
91623748 |
908 | } |
909 | printf("\n"); |
910 | } |
f278dcf4 |
911 | } |
912 | } |
913 | } |
914 | |
915 | qpos++; |
916 | } |
917 | |
918 | return s; |
919 | } |
920 | |
921 | static void free_sets(struct sets *s) |
922 | { |
923 | int i; |
924 | |
925 | freetree234(s->settree); |
926 | freetree234(s->outputtree); |
927 | for (i = 0; i < s->nsetlists; i++) |
928 | sfree(s->setlists[i]); |
929 | sfree(s->setlists); |
930 | for (i = 0; i < s->nnumberlists; i++) |
931 | sfree(s->numberlists[i]); |
932 | sfree(s->numberlists); |
933 | for (i = 0; i < s->noutputlists; i++) |
934 | sfree(s->outputlists[i]); |
935 | sfree(s->outputlists); |
936 | sfree(s); |
937 | } |
938 | |
939 | /* |
ed35622c |
940 | * Print a text formula for producing a given output. |
f278dcf4 |
941 | */ |
ed35622c |
942 | void print_recurse(struct sets *s, struct set *ss, int pathindex, int index, |
943 | int priority, int assoc, int child); |
944 | void print_recurse_inner(struct sets *s, struct set *ss, |
945 | struct ancestor *a, int pathindex, int index, |
946 | int priority, int assoc, int child) |
f278dcf4 |
947 | { |
ed35622c |
948 | if (a->prev && index != a->pr) { |
f278dcf4 |
949 | int pi; |
950 | |
951 | /* |
952 | * This number was passed straight down from this set's |
953 | * predecessor. Find its index in the previous set and |
954 | * recurse to there. |
955 | */ |
956 | pi = index; |
ed35622c |
957 | assert(pi != a->pr); |
958 | if (pi > a->pr) |
f278dcf4 |
959 | pi--; |
ed35622c |
960 | if (pi >= min(a->pa, a->pb)) { |
f278dcf4 |
961 | pi++; |
ed35622c |
962 | if (pi >= max(a->pa, a->pb)) |
f278dcf4 |
963 | pi++; |
964 | } |
ed35622c |
965 | print_recurse(s, a->prev, pathindex, pi, priority, assoc, child); |
966 | } else if (a->prev && index == a->pr && |
967 | s->ops[a->po]->display) { |
f278dcf4 |
968 | /* |
969 | * This number was created by a displayed operator in the |
970 | * transition from this set to its predecessor. Hence we |
971 | * write an open paren, then recurse into the first |
972 | * operand, then write the operator, then the second |
973 | * operand, and finally close the paren. |
974 | */ |
975 | char *op; |
976 | int parens, thispri, thisassoc; |
977 | |
978 | /* |
979 | * Determine whether we need parentheses. |
980 | */ |
ed35622c |
981 | thispri = s->ops[a->po]->priority; |
982 | thisassoc = s->ops[a->po]->assoc; |
f278dcf4 |
983 | parens = (thispri < priority || |
984 | (thispri == priority && (assoc & child))); |
985 | |
ed35622c |
986 | if (parens) |
987 | putchar('('); |
988 | |
688f45ad |
989 | if (s->ops[a->po]->flags & OPFLAG_UNARYPREFIX) |
ed35622c |
990 | for (op = s->ops[a->po]->text; *op; op++) |
991 | putchar(*op); |
992 | |
993 | print_recurse(s, a->prev, pathindex, a->pa, thispri, thisassoc, 1); |
994 | |
688f45ad |
995 | if (!(s->ops[a->po]->flags & OPFLAG_UNARYPREFIX)) |
ed35622c |
996 | for (op = s->ops[a->po]->text; *op; op++) |
997 | putchar(*op); |
998 | |
999 | if (!(s->ops[a->po]->flags & OPFLAG_UNARY)) |
1000 | print_recurse(s, a->prev, pathindex, a->pb, thispri, thisassoc, 2); |
1001 | |
1002 | if (parens) |
1003 | putchar(')'); |
f278dcf4 |
1004 | } else { |
1005 | /* |
1006 | * This number is either an original, or something formed |
1007 | * by a non-displayed operator (concatenation). Either way, |
1008 | * we display it as is. |
1009 | */ |
ed35622c |
1010 | printf("%d", ss->numbers[2*index]); |
f278dcf4 |
1011 | if (ss->numbers[2*index+1] != 1) |
ed35622c |
1012 | printf("/%d", ss->numbers[2*index+1]); |
1013 | } |
1014 | } |
1015 | void print_recurse(struct sets *s, struct set *ss, int pathindex, int index, |
1016 | int priority, int assoc, int child) |
1017 | { |
1018 | if (!ss->a.prev || pathindex < ss->a.prev->npaths) { |
1019 | print_recurse_inner(s, ss, &ss->a, pathindex, |
1020 | index, priority, assoc, child); |
1021 | } else { |
1022 | int i; |
1023 | pathindex -= ss->a.prev->npaths; |
1024 | for (i = 0; i < ss->nas; i++) { |
1025 | if (pathindex < ss->as[i].prev->npaths) { |
1026 | print_recurse_inner(s, ss, &ss->as[i], pathindex, |
1027 | index, priority, assoc, child); |
1028 | break; |
1029 | } |
1030 | pathindex -= ss->as[i].prev->npaths; |
f278dcf4 |
1031 | } |
1032 | } |
1033 | } |
ed35622c |
1034 | void print(int pathindex, struct sets *s, struct output *o) |
f278dcf4 |
1035 | { |
ed35622c |
1036 | print_recurse(s, o->set, pathindex, o->index, 0, 0, 0); |
f278dcf4 |
1037 | } |
1038 | |
ed35622c |
1039 | /* |
1040 | * gcc -g -O0 -o numgame numgame.c -I.. ../{malloc,tree234,nullfe}.c -lm |
1041 | */ |
f278dcf4 |
1042 | int main(int argc, char **argv) |
1043 | { |
1044 | int doing_opts = TRUE; |
1045 | const struct rules *rules = NULL; |
1046 | char *pname = argv[0]; |
1047 | int got_target = FALSE, target = 0; |
1048 | int numbers[10], nnumbers = 0; |
1049 | int verbose = FALSE; |
1050 | int pathcounts = FALSE; |
ed35622c |
1051 | int multiple = FALSE; |
91623748 |
1052 | int debug_bfs = FALSE; |
f278dcf4 |
1053 | |
1054 | struct output *o; |
1055 | struct sets *s; |
1056 | int i, start, limit; |
1057 | |
1058 | while (--argc) { |
1059 | char *p = *++argv; |
1060 | int c; |
1061 | |
1062 | if (doing_opts && *p == '-') { |
1063 | p++; |
1064 | |
1065 | if (!strcmp(p, "-")) { |
1066 | doing_opts = FALSE; |
1067 | continue; |
91623748 |
1068 | } else if (*p == '-') { |
1069 | p++; |
1070 | if (!strcmp(p, "debug-bfs")) { |
1071 | debug_bfs = TRUE; |
1072 | } else { |
1073 | fprintf(stderr, "%s: option '--%s' not recognised\n", |
1074 | pname, p); |
1075 | } |
f278dcf4 |
1076 | } else while (*p) switch (c = *p++) { |
1077 | case 'C': |
1078 | rules = &rules_countdown; |
1079 | break; |
1080 | case 'B': |
1081 | rules = &rules_3388; |
1082 | break; |
1083 | case 'D': |
1084 | rules = &rules_four4s; |
1085 | break; |
ed35622c |
1086 | case 'A': |
1087 | rules = &rules_anythinggoes; |
1088 | break; |
f278dcf4 |
1089 | case 'v': |
1090 | verbose = TRUE; |
1091 | break; |
1092 | case 'p': |
1093 | pathcounts = TRUE; |
1094 | break; |
ed35622c |
1095 | case 'm': |
1096 | multiple = TRUE; |
1097 | break; |
f278dcf4 |
1098 | case 't': |
1099 | { |
1100 | char *v; |
1101 | if (*p) { |
1102 | v = p; |
1103 | p = NULL; |
1104 | } else if (--argc) { |
1105 | v = *++argv; |
1106 | } else { |
1107 | fprintf(stderr, "%s: option '-%c' expects an" |
1108 | " argument\n", pname, c); |
1109 | return 1; |
1110 | } |
1111 | switch (c) { |
1112 | case 't': |
1113 | got_target = TRUE; |
1114 | target = atoi(v); |
1115 | break; |
1116 | } |
1117 | } |
1118 | break; |
1119 | default: |
1120 | fprintf(stderr, "%s: option '-%c' not" |
1121 | " recognised\n", pname, c); |
1122 | return 1; |
1123 | } |
1124 | } else { |
1125 | if (nnumbers >= lenof(numbers)) { |
1126 | fprintf(stderr, "%s: internal limit of %d numbers exceeded\n", |
1127 | pname, lenof(numbers)); |
1128 | return 1; |
1129 | } else { |
1130 | numbers[nnumbers++] = atoi(p); |
1131 | } |
1132 | } |
1133 | } |
1134 | |
1135 | if (!rules) { |
ed35622c |
1136 | fprintf(stderr, "%s: no rule set specified; use -C,-B,-D,-A\n", pname); |
f278dcf4 |
1137 | return 1; |
1138 | } |
1139 | |
1140 | if (!nnumbers) { |
1141 | fprintf(stderr, "%s: no input numbers specified\n", pname); |
1142 | return 1; |
1143 | } |
1144 | |
ed35622c |
1145 | s = do_search(nnumbers, numbers, rules, (got_target ? &target : NULL), |
91623748 |
1146 | debug_bfs, multiple); |
f278dcf4 |
1147 | |
1148 | if (got_target) { |
1149 | o = findrelpos234(s->outputtree, &target, outputfindcmp, |
1150 | REL234_LE, &start); |
1151 | if (!o) |
1152 | start = -1; |
1153 | o = findrelpos234(s->outputtree, &target, outputfindcmp, |
1154 | REL234_GE, &limit); |
1155 | if (!o) |
1156 | limit = -1; |
1157 | assert(start != -1 || limit != -1); |
1158 | if (start == -1) |
1159 | start = limit; |
1160 | else if (limit == -1) |
1161 | limit = start; |
1162 | limit++; |
1163 | } else { |
1164 | start = 0; |
1165 | limit = count234(s->outputtree); |
1166 | } |
1167 | |
1168 | for (i = start; i < limit; i++) { |
ed35622c |
1169 | char buf[256]; |
1170 | |
f278dcf4 |
1171 | o = index234(s->outputtree, i); |
1172 | |
ed35622c |
1173 | sprintf(buf, "%d", o->number); |
f278dcf4 |
1174 | |
1175 | if (pathcounts) |
ed35622c |
1176 | sprintf(buf + strlen(buf), " [%d]", o->npaths); |
f278dcf4 |
1177 | |
1178 | if (got_target || verbose) { |
ed35622c |
1179 | int j, npaths; |
f278dcf4 |
1180 | |
ed35622c |
1181 | if (multiple) |
1182 | npaths = o->npaths; |
1183 | else |
1184 | npaths = 1; |
1185 | |
1186 | for (j = 0; j < npaths; j++) { |
1187 | printf("%s = ", buf); |
1188 | print(j, s, o); |
1189 | putchar('\n'); |
1190 | } |
1191 | } else { |
1192 | printf("%s\n", buf); |
1193 | } |
f278dcf4 |
1194 | } |
1195 | |
1196 | free_sets(s); |
1197 | |
1198 | return 0; |
1199 | } |