Henry Ernest Dudeney/Puzzles and Curious Problems/106 - Digits and Primes/Solution

by : $106$

 * Digits and Primes

Solution
The solution given by is:

8 9  6 1   4 7     5     3 +    2 - 2 0 7

but we also have:

8 9  6 7   4 1     5     3 +    2 - 2 0 7

Proof
From the start we make the assumption that we can do this using primes less than $100$.

We have that $4$, $6$ and $8$ must appear in the tens column.

Hence we must use one element of each of the following sets:
 * $\set {41, 43, 47}$
 * $\set {61, 67}$
 * $\set {83, 89}$

By inspection we can see that $41$, $67$ and $89$ use up $6$ digits each, while the remaining digits are primes on their own.

There can be no sum lower than this.

We also note that $41 + 67 = 47 + 61$, giving a second solution.