Substitute each of the capital letters by a different digit from 0 to 9, such that each of the columns, each of the rows and each of the two main diagonals of this 3x3 grid have the common sum LRK. It is known that L is nonzero and, none of the numbers in the nine cells contains a leading zero.
CL RO OR
EK HS AA
KE KB SH
Note: LRK is equal to 100*L + 10*R + K.
1) B = 0, because every other letter either begins one of the 9 numbers or the sum
2) The sum must be over 100 and less than 300, so L is 1 or 2
3) CL + EK + KE = LRK, so L + E = 10 (based just on looking at the unit digits)
Since L is 1 or 2, E must be 8 or 9.
4) CL + RO + OR = CL + HS + SH = CL + EK + KE,
so RO + OR = HS + SH = EK + KE,
so R+O = E+K = H+S
since E is 8 or 9, this sum must be 9 or 11 or 12 or 13.
It can't be 10, because L + E = 10,
and the maximum is 13 because (4+5+6+7+8+9)/3 = 13
The only possibilities for E&K are
E K
- -
8 1
8 3
8 4
8 5
9 2
9 3
9 4
5) RO + HS + KB = KE + KB + SH = LRK
Based just on looking at the unit digits,
Since B = 0, O + S = K or 10 + K
and E + H = 10 + K (it can't equal K because E > K)
6) EK + HS + AA = LRK,
so based just on looking at the unit digits, S+A = 10
7) Let's work out the values for L, H, S & A, based on E & K
E K L=10-E H=10+K-E S=E+K-H A = 10-S
- - --------- ------------ ----------- ----------
8 1 2 3 6 4
8 3 2 5 6 4
8 4 2 6 6 4 X
8 5 2 7 6 4
9 2 1 3 8 2 X
9 3 1 4 8 2
9 4 1 5 8 2
7) Three of the rows are inconsistent, giving us just 4 possibilities. Let's add in values for O & R
E K L H S A O=10+K-S R=E+K-O
- - - - - - ------------ -----------
8 1 2 3 6 4 5 4 X
8 3 2 5 6 4 7 4 X
8 5 2 7 6 4 9 4 X
9 3 1 4 8 2 5 7
9 4 1 5 8 2 6 7
(Hmm. If I were more clever, I could have eliminated E = 8 earlier, By combining equations, I could have arrived at
R = 3E - 20 and A = 20-2E, so E cannot be 8)
8) At any rate, two of the rows are inconsistent, so the only possibilities are
E K L H S A O R B C
- - - - - - - - - -
9 3 1 4 8 2 5 7 0 6
9 4 1 5 8 2 6 7 0 3
9) Using the diagonals for the first time, we can rule out the first row based on
KE + HS + OR = LRK, so E + S + R = K (mod 10)
10) Therefore, 01234 56789 = BLACK HORSE, which hardly seems a coincidence.
This checks out, as
31 76 67
94 58 22
49 40 85 = 174
Edited on September 18, 2009, 5:41 pm
Edited on September 19, 2009, 1:23 am
Analytical Solution (and secret revealed)
