A perfect square is represented by BCCDEFGGB, where each of the letters is denoted by a distinct nonzero digit.
Furthermore:

• Each of BCCD and EFGGB is a perfect square.
• Each of EF and GGB is a perfect square.

Determine the square root of the number represented by BCCDEFGGB.

Note: Computer program solutions are welcome but a semi-analytic (calculator+p&p) method is preferred.

The requested square root is:  24035

The unique solution is:
577681225 5776 81225 81 225

-------------
sq2 = [i**2 for i in range(4,10)]
sq3 = [i**2 for i in range(10,32)]
sq4 = [i**2 for i in range(32,100)]
sq5 = [i**2 for i in range(100,317)]
sq9 = [i**2 for i in range(10000,31623)]

firstSols = []

for i in sq4:
    for j in sq5:
        x = int(str(i)+str(j))
        if x not in sq9:
            continue
        firstSols.append(x)
        strx = str(x)
        ef = int(strx[4:6])
        ggb = int(strx[6:9])
        if ef in sq2 and ggb in sq3:
            print(x, i, j, ef, ggb)

bccd = [i for i in sq4 if str(i)[1] == str(i)[2]]

ggbPossibles = []
for i in range(32,1000):
    stri3 = str(i**2)[-3:]
    if stri3[0] == stri3[1] and stri3[0] != stri3[2] :
        if int(stri3) not in ggbPossibles:
            ggbPossibles.append(int(stri3))

------- Of some interest --------
There are 16 solutions to the subproblem satisfying just the first condition, considering only BCCD and EFGGB:
102414400
129618225
152127556
160022500
193627225
230432400
270438025
313644100
360050625
409657600
462465025
518472900
577681225
592922500
640090000
705699225

The only possible values for GGB that perfect squares can end in:
[1, 4, 9, 116, 224, 225, 336, 441, 449, 556, 664, 776, 881, 884, 889, 996]  (G=0 for the single digit numbers)

Posted by Larry on 2023-04-10 14:40:57