Alice is playing with a new tetrahedral die. Each face has a different positive integer on it, but the numbers are peculiar, in that the numbers on the three exposed faces always sum to a perfect square.

Assuming that Alice’s tetrahedral die uses the smallest possible numbers, what are they?

Alice claps her hands in delight. 'How splendid it would be if all four sides also summed to a square!' she declares.

Is that possible?

(Adapted from Science 2.0)

The numbers with the smallest possible total are 1, 22, 41 and 58, adding to 122. The sum minus each possible hidden side is respectively, 121, 100, 81 and 64. The first several possibilities are:

                           total
1  22  41  58               122
9  34  57  78               178
7  34  59  103              203
14  41  66  89              210
5  34  61  130              230
12  41  68  116             237
1  32  88  136              257
3  34  63  159              259
10  41  70  145             266
26  57  86  113             282
1  34  65  190              290
4  68  97  124              293
6  39  99  151              295
8  41  72  176              297
24  57  88  144             313
33  66  97  126             322
2  37  130  157             326
6  41  74  209              330

The first few with square totals are:

120  177  232  432          961 *
65  128  248  648           1089 *
136  264  384  441          1225 *
73  144  408  744           1369 *
89  344  656  936           2025 *
176  344  504  1001         2025 *
97  192  552  1560          2401 *
97  552  720  1032          2401 *
192  376  801  1032         2401 *
99  291  979  1131          2500 *
99  475  651  1275          2500 *
208  600  873  1128         2809 *
216  321  424  2064         3025 *
321  424  624  1656         3025 *
440  545  1040  1224        3249 *
440  648  848  1313         3249 *
232  345  1272  1632        3481 *
232  672  777  1800         3481 *
345  456  880  1800         3481 *

DEFDBL A-Z
CLS
FOR tot = 1 TO 330
  FOR a = 1 TO tot / 4
    sq = tot - a
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
  FOR b = a + 1 TO (tot - a) / 3
    sq = tot - b
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
  FOR c = b + 1 TO (tot - a - b) / 2
    sq = tot - c
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
      d = tot - a - b - c
      sq = tot - d
      sr = INT(SQR(sq) + .5)
      IF sr * sr = sq AND d > c THEN
         sq = tot
         sr = INT(SQR(sq) + .5)
         IF sr * sr = sq THEN f$ = "*":  ELSE f$ = ""
         PRINT a; b; c; d, tot; f$
      END IF
    END IF
  NEXT c
    END IF
  NEXT b
    END IF
  NEXT a
NEXT tot

PRINT

FOR srtot = 1 TO 60
  tot = srtot * srtot
  FOR a = 1 TO tot / 4
    sq = tot - a
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
  FOR b = a + 1 TO (tot - a) / 3
    sq = tot - b
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
  FOR c = b + 1 TO (tot - a - b) / 2
    sq = tot - c
    sr = INT(SQR(sq) + .5)
    IF sr * sr = sq THEN
      d = tot - a - b - c
      sq = tot - d
      sr = INT(SQR(sq) + .5)
      IF sr * sr = sq AND d > c THEN
         sq = tot
         sr = INT(SQR(sq) + .5)
         IF sr * sr = sq THEN f$ = "*":  ELSE f$ = ""
         PRINT a; b; c; d, tot; f$
      END IF
    END IF
  NEXT c
    END IF
  NEXT b
    END IF
  NEXT a

NEXT

 

Posted by Charlie on 2013-07-22 18:37:13