2

This question is from the book 'Mathematical Circles', and it goes like this:

The secret service of The Federation intercepted a coded message from the Dominion which read: BLASE + LBSA = BASES. It is known that equal digits are coded with equal letters and different digits with different letters. Two giant computers came up with two different answers to the riddle. Is this possible or does one of them require repair?

The answer given is:

There is only one answer: 51286 + 1582 = 52868.
Hints:
L+L < 10
S+S $ \geq $ 10
If this did not hold, hundreds and units digits of $BASES$ would not be equal ($B \neq E$).

I was able to find the above solution but I am not able to see how there cannot be two answers. I know why $L+ L <10$ but I have trouble understanding why $S+S \geq 10$. As per my understanding, If: $E + A < 10$
Then $S + S > 10$. But why are we sure that this statement will hold if $E +A > 10$? I have also not arrived at any scenario where the contradiction $B=E$ arose.

Illusioner_
  • 453
Pumpkin_Star
  • 269
  • 1
    This is more a puzzle than a math question , although we can also apply logic instead of pure brute force. But I doubt we can solve this puzzle with logic alone and without any trial and error. – Peter Aug 19 '24 at 00:50

1 Answers1

3

$E+A\ge 10$ is not possible.

Looking at the units' digits, if $E+A\ge 10$ then $E+A=10+S$ with a carry to the tens' digits. Then

  • either $S+S+1=E$ in the tens digits with no carry to the hundreds' digits

    • in which case in the hundreds' digits, you need either $A+B =S$ (i.e. the impossible $B=E-10$) or $A+B=10+S$ (i.e. the prohibited $B=E$);

  • or $S+S+1=10+E$ in the tens' digits (with a carry), which would make $A+S=19$ (impossible with two single digits).

So $E+A< 10$, and thus you need $S+S \ge 10$ to avoid $B=E$ from the units' and hundreds' digits.

Henry
  • 169,616
  • Thanks a lot! This helps very much. Just one thing- Shouldn't it be S+S> 10 as otherwise E=0 and thus A=S? Also how do we proceed further with these two statements? – Pumpkin_Star Aug 19 '24 at 13:20
  • @Pumpkin_Star: I put $S+S \ge 10$ just to indicate there must be a carry. You can then go on to reject more specific values, as you do. But your original question said things like "S+S> or = 10". – Henry Aug 19 '24 at 13:36