## Teaser 2871: Five-card trick

**From The Sunday Times, 1st October 2017** [link]

I have five cards with a different digit from 1 to 5 on each. I shuffled them and placed them face-down in a row to form a concealed five-figure number. Then I invited each of my six nephews to choose a number less than fifty and they happened to choose six consecutive numbers. Then I explained that there would be a prize for anyone whose number was a divisor of the concealed number. No-one was certain to win but they were all in with a chance until I revealed the final digit of the number, which ruled out two of them from winning. Then I revealed the first digit and that ruled two more out. Then I revealed the whole number and just one nephew won a prize.

What was the concealed number?

[teaser2871]

## Jim Randell 11:48 am

on20 September 2019 Permalink |This is a similar puzzle to

Teaser 2891(also set by Stephen Hogg), but I think this one is easier to solve.This Python program runs in 52ms.

Run:[ @repl.it ]Solution:The concealed number is 15324.“no-one was certain to win”, eliminates divisors 1 and 3.

“but they all have a chance”, eliminates: 9, 10, 11, 18, 20, 22, 27, 30, 33, 36, 40, 41, 44, 45.

Which means the only set of six consecutive possible divisors remaining is: 12, 13, 14, 15, 16, 17.

Revealing the final digit eliminates 2 of the nephews, so the final digit must be 2 or 4 (if it were 1, 3, 5 at least the three nephews who had chosen even divisors would be eliminated), but if it were 2 only 15 would be eliminated. So the final digit is 4, and the nephews who chose 15 and 16 are eliminated.

Then the first digit is revealed, and this eliminates another two nephews. So the first digit is 1, which eliminates 13 and 17 as possible divisors (2 eliminates 12, 13, 14; 3 eliminates 13, 14, 17; 5 eliminates 17).

So the number is 1???4. Only 15324 has one winner – the nephew who chose 12. The other possibility is 13524, which has 12 and 14 as winners.

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