# Brain Teasers are Back!

There is a reason why such patterns arise. As it involves some ancient mathematics still in use today, I find it interesting.

Prime numbers are notoriously difficult to identify directly. In the 3rd century BC, Greek scholar Eratosthenes realised it was easier to identify all the non-primes (composites), so what’s left would be a list of primes. Very astute!

His approach was to write out all the integers from 2 to n, then cross out every second integer (even numbers) starting from 4. Next, cross out every third integer starting from 9. Then every fifth integer starting from 25 etc. It’s known as the Sieve of Eratosthenes.

It does have some inefficiencies though. It is rather laborious to write out all the numbers only to cross out half of them at the next step. This can be considered as a cyclical pattern (of composites) repeating every two integers. There is a further cyclical pattern for multiples of 3, repeating every 2x3 integers. This can be extended for multiples of 5, repeating every 2x3x5 =30 integers etc.

The approach is known as Wheel Factorisation and can be seen visually (diagram from Wikipedia):

As the yellow regions are automatically composites they can be ignored which reduces the size of the problem (primes can only occur in 8 of the 30 spokes).

The reason why Don needed to exclude 43-50 and 73-80 is because these ranges include multiples of 7 that are not multiples of 2, 3 or 5. Namely 49 and 77.

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Really interesting Ravvie, thank you.

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As SteveD says Ravvie, really interesting. Thank you.

Sequences …
Another straight forward sequence to continue …

8, 27, 125, 343, 1331 … ? ? ?

how about 2197, 4193, 6859 … (cube of prime numbers)?

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Well done Steve, well done.

Hopefully, not too difficult

Pointless, but I think it’s around 228 pence short of £20

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Well done Steve.

You had me confused for a moment or two. But spot-on !

Many thanks Steve for your disguised response.

I thinks others on the Forum could still provide the solution. Either by working through the original teaser - as you did - or by solving your response, which is itself a neat teaser

I haven’t looked here for a while, but this one seems easier than many…

(The two digits added followed by them multiplied)

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Hi IB, yes, some of the probability Teasers were getting a bit tough, especially when young Ravvie adds an ‘extension’ to an otherwise simple probability/sequence/etc …

Likewise, Willy set a couple of “impossible” teasers a few weeks back, which have only been partially resolved.

Anyhow, well done. And a neat explanation of the solution. Hope you enjoyed.

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I prefer infinite improbability!

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Okay, I will see what I can do

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Is that definite, or are you just sure ?

Most probably perhaps!

Odd one out ?

Which of the following numbers is the odd one out ?

196; 169; 361; 81; 518; 324; 64

Well, I could say 64 as it hasn’t got a winking emoji after it…
But I will say 518 because the Odyssey song applies to the other numbers but not 518

My eyes just glazed over

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Hi Steve, Hi Mike

518 is correct, and no doubt you know the reason. Well done, Steve, and well done Mike.

However, you have given me a real Brain Teaser with your reference to 'the Odyssey song ’
I have no idea what this is, despite Googling those words 'the Odyssey song ’ and reading the lengthy lyrics !!

Also somewhat puzzling, is the reference to ‘winking emoji’.
My version of the teaser has a string of numbers, each separated by a semi-colon.
Not an emoji in sight !

Baffled