It is a mathematical puzzle, where the numbers are represented by letters or symbols. Each letter represents a unique digit. The goal is to find the digits such that a given mathematical equation is verified.

In general, there are few variables in form of letters which is to be assigned numeric values in the range of 0 to 9, such that the given equation hold true value.

Example:

IS + WT + SPM = BEIT

One of the possible solution is I=3 S=4 W=8 T=5 S=9 P=1 M=6 B=1 E=0 as,

34 + 85 + 916 = 1035

The above e.g. has multiple solutions. Ideally, a good Crypto-Arithmetic puzzle must have only one solution.

How to solve the problem?

We follow the classical three-stage method.

Stage1: Describe

1. In the describe stage we will explain problem and goal in natural language.

2. For a complete problem description we need to answer following three questions:

3. What is the goal of the problem?

In this e.g., the goal is to replace letters by digits such that the sum is IS+WT+SPM=BEIT is verified.

1. Are there any unknowns or decision variables?

2. What are the constraints?

The obvious constraint is the sum that has to be verified and all the variables must have different values in a feasible solution. But as we observe the given equation, we notice that, there are other implicit constraints as well.

1. It is implicit that letters I, W, S and B cannot represent digit 0, as they are the first digit of a number.

2. There are 9 distinct letters, so we need at least9 different digits in the answer.

3. By observation, we have B=1, Consider three auxiliary variables, X,Y,Z. Hence we can write the constraint equations as,

S +T+M=10X+T

I+W+P+X=10Y+I

S+Y=10+E

Solve:

We got the quadratic equations representing the problem, we can solve them simultaneously to get the solution.

Example 2:

T W O

+T W O

F O U R

E.g. Setting F 1 O=4

Trick: introduce auxiliary variables X,Y