# Express 2222 as a sum of 3 powers in 2 Different Pan Digital Expressions

Solve for the digits in this Unique set of Palindrome- Pan Digital Expressions:

$$2222$$ = $$(A/B)^C$$ + $$(D/E)^F$$ + $$(G/H)^I$$

$$2222$$ = $$(A/B)^I$$ + $$(D/E)^F$$ + $$(G/H)^C$$

Where, A,B,C,D,E,F,G,H,I are distinct digits representing digits 1 to 9.

## 1 Answer

The solutions are

$$2222=(9/3)^7+(8/4)^5+(6/2)^1=(9/3)^1+(8/4)^5+(6/2)^7$$

Explanation (not full and without a proof of uniqueness)

First, find a power of a single-digit integer (with single-digit exponent) slightly less than 2222. Of course, we have $$3^7=2187$$. The remainder - $$2222-2187=35$$ can be represented as the sum of 2 powers - $$35=2^5+3^1$$ in the easiest way. So, we have $$2222=3^7+2^5+3^1$$. This is a "good" representation of $$2222$$, since it contains 2 powers of 3, so we can probably find both solutions, and the power bases are small enough (2 and 3) to be represented as the quotient of 2 digits in several ways. Now, arranging the remaining digits to get the desired quotients is trivial task.