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

Puzzles and Curious Problems by Henry Ernest Dudeney: $106$

Using the $9$ digits once, and once only,

can you find prime numbers that will add up to the smallest total possible?

The solution given by Dudeney is:

but we also have:

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.

$\blacksquare$