The letters denote a different base ten digit from 0 to 9. No number can contain any leading zero.
It is known that both the relationships are simultaneously satisfied:
1) (AAAU)2= ALUMINE
2) (A+A+A+U)2= A+L+U+M+I+N+E
What is the number denoted by ALUMINE?
Looking solely at the first alphametic.
AAAU is a 4-digit number and ALUMINE is a 7-digit number. Then AAAU < sqrt(10,000,000)=3162.3
This limits A to 1, 2, or 3.
AAAU is close to AAAA, so let's test the repdigits 1111, 2222, and 3333: 1111^2=1234321, 2222^2=4937284, 3333^2=11108889
The only square that shares the same leading digit is 1111^2=1234321.
Then this suggests ALU=123, which then makes AAAU=1113.
Verify this as our solution: 1113^2=1238769 and (1+1+1+3)^2=36=1+2+3+8+7+6+9.
The number denoted by ALUMINE is 1238769.
Analytic Solution
