# Letters and Numbers

Every letter is a different value from 1-26

1. A x A = L + G
2. C x C = U
3. J x H = P + R
4. W - K = N
5. G + L = E + Z
6. S + J = P - Y
7. E x C = Z x B
8. D + U = Z + E
9. G x F = D + U
10. O + X = V
11. S x K = Z + E
12. K x R = M
13. D + U = R + P
14. Q is higher than T
15. O is higher than I

Can you work out their values?

• Hi! Welcome to Puzzling. Is this your own puzzle? If not, we need to have proper attribution for puzzles that you didn't create; this helps prevent plagiarism on our site. Thanks, and again, welcome! Commented Jan 21 at 10:05
• Hi, I wrote it, sorry i didnt know i have to mention that Commented Jan 21 at 10:15
• @LizardofOz you don't have to mention that it is your own puzzle; if it is not yours you have to provide proper attribution.
– Sny
Commented Jan 21 at 13:56
• It is still safer to mention it is one's own puzzle if it looks like it could have been taken from somewhere else, or it might get closed. Commented Jan 21 at 18:19
• A and C can only be 3, 4 or 5. Commented Jan 21 at 18:22

The key to this puzzle is that

Lines 1, 3, 5, 8, 9, 11 and 13 all have the same value: a square with at least 7 divisors greater than 1. These are A,F,G,H,J,K and S. The only possible value for this is square is 36, so we have A = 6 and the other six are 2, 3, 4, 9, 12, and 18.

We can proceed with:

C x C = U must be 5 x 5 = 25, as C is not a factor of 36.
U + D = 36 gives us D = 11.
F x G = G + L = 36, and this can only be 3 x 12 = 12 + 24.
R + P = 36 gives R > 9 and K x R = M gives us K = 2
S x K = 36 gives us S = 18
E x C = Z x B with Z + E = 36 gives us 36 / E = (5 + B) / B.
Trial and error finds E = 21 and B = 7

Further trial and error and a process of elimination complete the solution:

 A =  6  |  N = 17
B =  7  |  O = 14
C =  5  |  P = 23
D = 11  |  Q = 20
E = 21  |  R = 13
F =  3  |  S = 18
G = 12  |  T = 16
H =  9  |  U = 25
I = 10  |  V = 22
J =  4  |  W = 19
K =  2  |  X =  8
L = 24  |  Y =  1
M = 26  |  Z = 15

• perfectly done, nice job Commented Jan 21 at 20:28

Looks like a correct answer got posted while I was working on this, but the following identifies all the values without trial and error:

First observe that there are lots of common elements in (1), (3), (5), (8), (9), (11) and (13).
Combining them we have A*A = G*F = S*K = J*H = L+G = E+Z = D+U = R+P.
The sums must all be between 3 and 51, so we need a square between 4 and 49 that can be factored four different ways.
The only possibility is 36 = 6*6 = 4*9 = 3*12 = 2*18.
So A = 6.

All the letters in the sums must be at least 10, but cannot be 18, and as G appears in both a sum and a product we know G = 12.
And that means F = 3 and L = 24.

Now look at (12): K x R = M
We know R >= 10 from the sums to 36, so K <= 2. But K=1 gives R=M.
So K = 2.
From the sums to 36, we know R >= 10, so 10 <= R <= 13, and 20 <= M <= 26.
And from the products to 36, we have S = 18, leaving J and H as 4 and 9 (but we don't know which way round yet).

Now look at (6): S + J = P - Y
We know S = 18, and J = 4 or 9. But the difference can be at most 25, so J = 4 .

So we have 22 = P - Y.
So Y <= 4, and the only free option is Y = 1.
Leaving P = 23.

And from the sums and products to 36, that means R = 13 and H = 9.
And from (12), M = 26.

Now look at (2): C x C = U.
The square gives us 2 <= C <= 5, and the only remaining option is C = 5.
So U = 25. And from the sums to 36, that means D = 11.

Now look at (7) - E x C = Z x B
We know C = 5, so Z*B must be a multiple of 5.
We also know E and Z sum to 36, and with several values taken, both must be between 14 and 22.
Rearranging we have E/Z = B/5.
With all lower values taken, we know B >= 7, and that also means E > Z.

If B >= 10 we would have E/Z >= 2, which is impossible given the possible values.
So 7 <= B <= 9

And that means Z must be a multiple of 5, and the only possibility is Z = 15.
So E = 21. And B = 7.

Now look at (10): O + X = V
With most values taken, there is only one possibility here: 8 + 14 = 22.
So V = 22.
And O,X are 8,14 in some order.

Looking at (15): O is higher than I
O cannot be 8, as there are no lower values free for I.
So O = 14 and X = 8.
And I = 10 (The only lower value available.)

Now look at (4): W - 2 = N
There is only one option among the remaining values: 19 - 2 = 17.
So W = 19 and N = 17

And finally (14): Q is higher than T makes Q = 20 and T = 16

1   Y
2   K
3   F
4   J
5   C
6   A
7   B
8   X
9   H
10  I
11  D
12  G
13  R
14  O
15  Z
16  T
17  N
18  S
19  W
20  Q
21  E
22  V
23  P
24  L
25  U
26  M