Find a base ten perfect square of 12 digits formed from the concatenation of two base ten perfect squares, one having 4 digits and the other 8 digits. None of the three perfect squares can begin with a zero.

1024 || 57608100 =  102457608100
1521 || 97515625 =  152197515625
2401 || 98010000 =  240198010000
3249 || 57002500 =  324957002500
3844 || 24800400 =  384424800400
3969 || 25200400 =  396925200400
6241 || 77422401 =  624177422401
8649 || 55800900 =  864955800900
9801 || 99002500 =  980199002500

None were found with the 8-digit square concatenated before the 4-digit square.

Perhaps the "best" one is 152197515625, as it not only has no leading zeros, but no zeros at all.

DEFDBL A-Z
a0 = -INT(-SQR(1000)): a1 = INT(SQR(9999))
b0 = -INT(-SQR(10000000)): b1 = INT(SQR(99999999))
CLS
FOR a = a0 TO a1
  asq = a * a
FOR b = b0 TO b1
  bsq = b * b
  ck1 = 100000000 * asq + bsq
  ck2 = 10000 * bsq + asq
  sr1 = INT(SQR(ck1) + .5)
  sr2 = INT(SQR(ck2) + .5)
  IF sr1 * sr1 = ck1 THEN PRINT ck1
  IF sr2 * sr2 = ck2 THEN PRINT "             "; ck2
NEXT
NEXT

 

Posted by Charlie on 2011-09-25 16:24:27