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 Square Tetrahedral Dice (Posted on 2013-07-22)

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)

 See The Solution Submitted by broll No Rating

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 computer solution Comment 1 of 1

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:

`                           total1  22  41  58               1229  34  57  78               1787  34  59  103              20314  41  66  89              2105  34  61  130              23012  41  68  116             2371  32  88  136              2573  34  63  159              25910  41  70  145             26626  57  86  113             2821  34  65  190              2904  68  97  124              2936  39  99  151              2958  41  72  176              29724  57  88  144             31333  66  97  126             3222  37  130  157             3266  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
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