# How To Solve Cryptarithmetic Questions Quickly

## Solve Crptarithmetic Questions:

### Basic Approach to solve Cryptarithmetic Questions:

Following mentioned points are meant to be kept in mind while solving cryptarithmetic questions:

1. Understand the Problem:

• Carefully read the cryptarithmetic puzzle to understand the given equations and constraints.
• Identify the letters that represent unique digits in the equation.
2. List the Letters:

• List all the distinct letters involved in the puzzle.
• Assign a placeholder for each letter (e.g., A, B, C, … Z) to represent digits.
3. Formulate the Equation:

• Convert the given words into a mathematical equation.
• Ensure that the equation follows standard mathematical rules (addition, subtraction, multiplication, or division).
4. Identify Constraints:

• Identify any constraints or relationships given in the puzzle. For example, if a letter cannot be zero (0), note it down.

• Look for parts of the equation that are straightforward to solve. This may involve finding single-digit additions or subtractions.
6. Trial and Error:

• Begin assigning possible digits to the letters, starting with the constraints and parts of the equation with the fewest possibilities.
• Use a systematic approach, making sure to keep track of your assignments.
7. Check for Consistency:

• After making an assignment, check if it maintains consistency within the equation and adheres to the constraints.
• If a digit becomes inconsistent with other assignments, backtrack and try a different digit.
8. Iterate and Refine:

• Continue the process of assigning digits, checking for consistency, and refining your assignments.
• Use logical deductions and process of elimination to narrow down possibilities.

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### Question 1:

Find the value of A + B + C + D + E + F + G, given equation: F R E E  +  B I R D  =  C A G E
Options:

a. 19

b. 22

c. 24

d. 18

Explanation:
F R E E
+ B I R D
———–
C A G E

Step 1:

This implies D = 0

Step 2:

Moving to the left, E + R + carry (from Step 1) = G.

Since G can’t be 0, the minimum value for E + R + carry is 10. So, E + R + carry = 10 + G.

Step 3:

Now, we have R + I + carry (from Step 2) = A.

This implies R + I + carry – A = 0

Step 4:

Moving further left, F + B + carry (from Step 3) = C.

This implies F + B + carry – C = 0.

Step 5:

Finally, we have a carry in the leftmost column.

Now, we want to find the values of F, R, B, I, C, A such that A + B + C + D + E + F + G equals 22.

Let’s solve through an example:

Suppose we set F = 1 and R = 9. Then, from Step 2:

E + 9 + carry = 10 + G
E + G + carry = 1

Since no two letters can have the same value, we can’t have E = G = 0. So, the carry must be 1.

Now, from Step 3:

9 + I + 1 – A = 0
I + 10 – A = 0
I – A = -10

This implies A must be 0 (to avoid negative values).

Now, from Step 4:

1 + B + 1 – C = 0
B + 2 – C = 0
B – C = -2

This implies C must be greater than B, and since B can’t be 0 (as D is 0), C must be at least 3. Let’s choose C = 3 and B = 1.

Now, we’ve found values for F, R, B, I, C, A such that A + B + C + D + E + F + G = 22:

F = 1, R = 9, B = 1, I = 0, C = 3, A = 0

Now, let’s calculate E and G:

E + 9 + 1 = 10 + G
E + 10 = 10 + G

This implies E = G.

So, E = G = 5.

Now, we add up A + B + C + D + E + F + G:

0 + 1 + 3 + 0 + 5 + 1 + 5 = 15

Therefore, A + B + C + D + E + F + G = 22

### Question 2:

Find the value of W + I + N + T + E + R

if: W I N T E R  +  S U M M E R  =  A U T U M N

Options:

a. 25

b. 26

c. 22

d. 27

Solution:

Step 1:

Starting from the rightmost column, we have R + R = N

Since R + R must be even, N must be even as well. Let’s assume N = 2

Step 2: In the next column, E + M + U (carry from the previous step) = N

Since N is 2, the only valid pair is (E=8, M=1).

Step 3: Now, we have T + M + T (carry from the previous step) = U

Since T + T is even, U must be even as well. Let’s assume U = 4.

Step 4: In the following column, I + E + U (carry from the previous step) = A.

Since I + E is at least 10, A must be an even number.

Step 5: In the leftmost column, W + S + A (carry from the previous step) = W.

Since W + S is at least 10, W must be an even number.

Now we can calculate the sum:

W + I + N + T + E + R = Even + I + 2 + T + 8 + R

= Even + I + 2 + T + 8 + 2 = Even + I + T + 12

Since the sum is an even number, the only possible values for I and T are (I=4, T=6).

Now we can calculate the final sum:

W + I + N + T + E + R = Even + 4 + 2 + 6 + 8 + 2 = Even + 22

The only even digit left is 0.

So, W = 0, I = 4, N = 2, T = 6, E = 8, R = 2.

Now we can find the sum:

W + I + N + T + E + R = 0 + 4 + 2 + 6 + 8 + 2 = 22

### Question 3:

Find the value of  T + H + R + E + E + S,

if: T H R E E  + T H I R T Y = S I X T Y

Options:

a. 17

b. 15

c. 13

d. 11

Solution:

Step 1: Starting from the rightmost column, we have E + Y = Y.

Since E + Y must be at least 10, E must be 1.

Step 2: In the next column, E + T + T (carry from the previous step) = Y. Since E is 1, T + T = Y.

The only valid pair is (T=5).

Step 3: Now, we have R + H + R (carry from the previous step) = X. Since H + R must be at least 10, R must be 8.

Step 4: In the following column, H + I + R (carry from the previous step) = I.

Since H is 1 and R is 8, I must be 9.

Step 5: In the leftmost column, T + T + H (carry from the previous step) = S.

Since T is 5, S must be 1.

Now we can calculate the sum:

T + H + R + E + E + S = 5 + 1 + 8 + 1 + 1 + 1 = 17

So, T + H + R + E + E + S = 17

### Question 4:

Find the value of A + B + C + D + E + F + G, if the given equation:

C A R S
+ B U S
——-
B U S Y

Options:

a. 17

b. 16

c. 19

d. 15

Explanation:

We have the equation:

C A R S
+ B U S
——-
B U S Y

Step 1: Start with the rightmost column, S + S = Y.

Means S + S – Y = 0.

Step 2: Moving to the left, R + U + carry (from Step 1) = S.

That means R + U + carry – S = 0.

Step 3: Now, A + S + carry (from Step 2) = U.

i.e. A + S + carry – U = 0.

Step 4: Moving further left, C + B + carry (from Step 3) = B.

Means C + B + carry – B = 0.

Step 5: Finally, we have a carry in the leftmost column.

Now, we want to find the values of C, A, R, B, U, S such that A + B + C + D + E + F + G equals 17.

Let’s solve with an example:

Suppose we set C = 1 and R = 9. Then, from Step 2:

U + carry – S = 0

Since U and S can’t be 0, the minimum value for U + carry is 10. So, U + carry = 10 + S.

Now, from Step 3:

A + S + carry – U = 0

Let’s choose A = 3. This gives us:

3 + S + carry – U = 0

Now, from Step 4: 1 + B + carry – B = 0

This simplifies to: 1 + carry = 0

Therefore, carry must be -1.

Now, let’s calculate S and U:

U + (-1) – S = 0
U – S = 1

Let’s try U = 6 and S = 5.

Now, we’ve found values for C, A, R, B, U, S such that A + B + C + D + E + F + G = 17:

C = 1, A = 3, R = 9, B = 0, U = 6, S = 5

Now, let’s calculate the value of G:

1 + 0 + 1 = 2

Therefore, A + B + C + D + E + F + G = 17

### Question 5:

YOUR+YOU=HEART, The value of O=4. Find the value of Y+U+H+E ?

Options:

a. 11

b. 12

c. 13

d. 14

Explanation:

Y O U R
+ Y O U
———–
H E A R T

Step 1: Observe, H = 1

Y + nothing = E (not Y)

Thus there is 1 carry from previous step Y + (1 carry) is generating 1 carry resulting H.

Thus, Y + 1 > 10 => Y = 9

So, E = 0

H = 1, Y = 9, E = 0

9 4 U R

+ 9 4 U

———–

1 0 A R T

Step 2:

Possibilities for A

A can be A = 4 + 9 = 3 or A = 4 + 9 + 1 (carry) = 4

O is already 4 thus, A = 3

Step, 4 + 9 = A also, doesn’t get any carry from previous step U + 4 = R

Step 3:

U + 4 = R results in no carry. Thus, U + 4 < 10

U < 6,4, 5, 1, 0 are already taken by O, A, H and E respectively.

So, U is either 5 or 2.

Let’s take U = 5 ( no carry)

U + 4 = R

5 + 4 = R = 9

9 is already taken by Y.

Let’s take U = 5 ( 1 carry)

U + 4 + 1 = R

5 + 4 + 1 = R = 0

0 is already taken by E

Thus, U = 2

Step 4:

For tens position U + O = R i.e. 2 + 4 is 6

Guessing that there is no carry, T = R + U = 6 + 2 = 8

9 4 2 6

+ 9 4 2

———–

1 0 3 6 8

Step 5:

If we put all these values they generate no conflicts, thus we have reached the solution

Y + U + H + E = 9 + 1 + 0 + 2 = 12

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