Cryptarithms - Math Tutorial

Cryptarithms

A cryptarithm is a type of mathematical puzzle in which the digits are replaced by symbols (typically letters of the alphabet).

For example:

9567 + 1085 = 10652
Can be represented by:
abcd + efgb = efcbh

Cryptarithm Rules

• Numbers must not start with a zero.
• Each letter represents a unique digit.
• The solution is unique (unless otherwise stated).

 Sample 1:

Find the solution for ABCDE x 4 = EDCBA.

Step 1: Solving A
Possible A: 1,2

Why: 3-9 when multiply by 4 will produce 6 digit numbers.

Answer: A=2

Why: A must be even number in order to be divisible by 4.

New Equation: 2BCDE x 4 = EDCB2

Step 2: Solving E
Possible E: 3,8

Why: 3 and 8 are the only values that has a product ending of 2 when multiply by 4

Answer: E=8

Why: Only A=8 can satisfy the new equation from step 1.

New Equation: 2BCD8 x 4 = 8DCB2

Step 3: Solving B
Possible B: 0,1,2

Why: When 4 is multiplied by b, (0,1, and 2) are the only possible numbers in order to satisfy the new equation from step 2. (if 3, 4 x 3 = 12, 2 + 1(carry from 12) = 3 which is not equal to 8

Answer: B=1

Why: 8DCB2 must be divisible by 4. The divisibility rule for 4 is that the last 2 digits must be divisible by 4. Only B=1 can satisfy this statement which happened to be 12.

New Equation: 21CD8 x 4 = 8DC12

Step 4: Solving D
Possible D: 2,7

Why: The product of 4 and D plus 3 (carry from 4 x 8 = 32) must end in 1 to satisfy the new equation from step 3.

Answer: D=7

Why: D must be greater the or equal to 4. It will be prove when 4 is multiplied by B=1.

New Equation: 21C78 x 4 = 87C12

Step 5: Solving C
Possible C: 7,8,9

Why: 4 x C + 3 (carry from 4 x 7 + 3 = 31) must be greater than 30 - 39 in order to carry the 3 to be added to B to get 7.

Answer: D=9

Why: 9 is the only number that satisfy the equation.

Final Answer:   21978 x 4 = 87912
 
 
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