###
The CryptArithmetic Problem's Solution

**What was the problem?**
**L**et's first recall the given equation.
BASE +

BALL

---------

GAMES

----------
We are assuming repeating the numbers are not allowed.
Let's first take** last** **2 di****gits** operation into consideration i.e. **SE + LL ****= ES** or 1ES (carry in 2 digit operation can't exceed 1). For a moment, let's assume no carry generated.
10S + E + 10L + L = 10E + S .....(1)
**9 (E - S) = 11L**
To satisfy this equation L must be 9 and **(E**** -**** S)** must be equal to **11**. But difference between 2 digits can't exceed **9**. Hence, SE + LL must have generated carry.So rewriting (1),
10S + E + 10L + L = 100 + 10E + S
**9 (E - S) + 100 = 11L**
Now if [9 (E - S)] exceeds 99 then L must be greater than 9. But L must be digit from 0 to 9. Hence, [9 (E - S)] must be** negative** bringing down LHS below 100. Only value of** E**** ****- S** to satisfy the given condition is **-5** with** L ****= 5.** Or we can say,** S - E = 5.**
Now, possible pairs for SE are (9,4), (8,3), (7,2), (6,1), (5,0). Out of these only **(8,3****)** is pair that makes equation SE + LL = SE + 55 = 1ES i.e. 83 + 55 = 138. Hence, **S**** = 8** and **E = 3.**
Replacing letters with numbers that we have got so far.
1

---------
BA**83** +

BA**55**

---------

GAM**38**

----------
Now,** M ****= 2A + 1.** Hence, M must be odd number that could be any one among **1,****7,9** (since 3 and 5 already used for E and L respectively).
If M = 1, then A = 0 and B must be 5. But L = 5 hence M can't be 1
If M = 7, then A = 3 or A = 8. If A = 3 then B = 1.5 and that's not valid digit. And if A = 8 then it generates carry 1 and 2B + 1 = 8 again leaves B = 3.5 - not a perfect digit.
If** ****M ****= 9****,** then** ****A ****= 4** (A = 9 not possible as M = 9) and **B** must be **7** with carry **G ****= 1.**Hence for first 2 digits we have 74 + 74 + 1 = 149.
Finally, rewriting the entire equation with numbers replacing digits as -
1
---------
**74****83** +

**74****55**

---------

**149****38**

----------
So numbers for letters are **S = 8, E = 3, L = 5, A = 4, B = 7, M = 9 and G = 1.**
** **

## No comments:

Post a Comment