## Teaser 2964: “Bingo a go-go” lingo a no-go

**From The Sunday Times, 14th July 2019** [link]

My rest home’s Bingo set uses numbers 1 to 99. To win, nine numbers on your game card must be called. Our caller, not knowing “bingo-lingo”, says “Number 1, total factors 1”, “Number 11, total factors 2” and “Number 30, total factors 8”, etc.

Yesterday, in one game, my hearing aid howled whenever a call started. I missed each number, but heard each “total factors” value. Fortunately, after just nine calls I shouted “HOUSE!” certain that I’d won.

I told my daughter how many different “factor” values I’d heard, but didn’t say what any of the values were. Knowing that I had won after nine calls, she could then be sure about some (fewer than nine) of my winning numbers.

Find, in ascending order, the numbers that she could be sure about.

[teaser2964]

## Jim Randell 6:43 pm

on12 July 2019 Permalink |(See also:

Enigma 1004for another puzzle that involves counting divisors).This Python program groups the numbers from 1 to 99 into collections that have the same number of divisors, and then looks for groups of those collections that give a set of exactly 9 numbers, and these groups are recorded by the size of the group.

We can then find groups of the same size that have a non-empty set of fewer than 9 numbers in common, and this gives our solution.

It runs in 78ms.

Run:[ @repl.it ]Solution:The 7 numbers she could be sure about are: 1, 4, 9, 25, 36, 49, 64.The daughter was told that 5 different “factor” values were heard, so she knows the divisors are either:

So the solution is given by the subset of the numbers that have 1, 3, 7, or 9 divisors.

1 is the only number with 1 divisor.

4, 9, 25, 49 (the squares of primes) have 3 divisors.

64 (a prime to the power of 6) has 7 divisors

36 (the square of the product of two different primes) has 9 divisors.

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