# Product equality and sum of squares equality puzzle

• K Sengupta
In summary, the Product Equality and Sum of Squares Equality Puzzle is a mathematical problem that involves finding two numbers whose product is equal to their sum plus one. It can be solved using algebraic equations, with the possible solutions being (x,y) = (2,2) or (x,y) = (0,0). While there are other methods to solve the puzzle, the algebraic method is the most widely used. While the puzzle itself may not have practical applications, the equations used to solve it are commonly used in various fields. There are also variations of the puzzle that involve finding different relationships between two numbers.
K Sengupta
Substitute each of the capital letters by a different digit from 0 to 9 to satisfy this set of cryptarithmetic relationships. None of the numbers can contain any leading zero.

ABCD*EF=GHJB*KE, and:

(EH)2 + (KC)2 = (KH)2

abcd*ef=ghjb*ke
9807*14=6538*21

(eh)2 + (kc)2 = (kh)2
(15)2 + (20)2 = (25)2

## 1. What is the Product Equality and Sum of Squares Equality Puzzle?

The Product Equality and Sum of Squares Equality Puzzle is a mathematical problem that involves finding two numbers that have a specific relationship. Specifically, the puzzle requires finding two numbers whose product is equal to their sum plus one.

## 2. How do you solve the Product Equality and Sum of Squares Equality Puzzle?

The puzzle can be solved using algebraic equations. Let x and y be the two numbers we are trying to find. We can set up the following equation: xy = x + y + 1. By rearranging the terms, we get xy - x - y = 1. We can then factor the left side to get (x-1)(y-1) = 1. This means that (x-1) and (y-1) are factors of 1. The only possible values for (x-1) and (y-1) are 1 and 1, or -1 and -1. Therefore, the two possible solutions are (x,y) = (2,2) or (x,y) = (0,0).

## 3. Are there any other ways to solve the Product Equality and Sum of Squares Equality Puzzle?

Yes, there are other methods to solve the puzzle such as using geometric shapes and patterns. However, the algebraic method is the most widely used and efficient way to solve the puzzle.

## 4. Is the Product Equality and Sum of Squares Equality Puzzle used in real-world applications?

While the puzzle itself may not have practical applications, the algebraic equations used to solve it are commonly used in various fields such as engineering, finance, and physics.

## 5. Are there any variations of the Product Equality and Sum of Squares Equality Puzzle?

Yes, there are variations of the puzzle that involve finding different relationships between two numbers, such as their product being equal to their sum minus one. These puzzles can be solved using similar algebraic methods.

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