A three-digit number grows by if we exchange the second and third digits and grows by if we exchange the first and second digits. By how much will it grow if we exchange the first and third digits?

Source: NCTM Mathematics Teacher, January 2006

**SOLUTION**

Suppose we start with .

Exchanging the digits and yields the new number

the number grows by

Exchanging the digits and yields the new number

the number grow by

It looks like the digits have to be consecutive in order to have a smaller growth.

So let’s try .

Exchanging the digits and yields the new number

the number grows by as before.

Exchanging the digits and yields the new number

the number grows by as expected.

Exchanging the digits and yields the new number

the number grows by .

The answer seems to be , but we need to prove it.

Suppose represent the three digits of a number.

Step 1: becomes

Simplify

Step 2: becomes

Simplify

Step 3: becomes

Adding Eq. to Eq.

————–

The growth equals

**Answer**: