Henry Ernest Dudeney/Puzzles and Curious Problems/106 - Digits and Primes/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $106$
- Digits and Primes
- Using the $9$ digits once, and once only,
- can you find prime numbers that will add up to the smallest total possible?
Solution
The solution given by Dudeney 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.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $106$. -- Digits and Primes
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $134$. Digits and Primes