h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

A quick arithmetic brainteaser for you today.

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I can order McDonalds Chicken McNuggets in boxes of 6, 9 or 20.

What's the largest number of McNuggets I can't order exactly?

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It doesn't exist.

That looks like a pretty good answer, Bel. (The largest prime found to date is nearly 13 million digits long, and there must be even larger ones.)

But, Icy might just have something up his sleeve.

We probably assume our world has literally endless supplies of nugget boxes exist

I was wondering when the first mathematic flavoured quiz would turn up

I'm surprised that somebody hasn't suggested 42!

In fact it is 43.

Explanation to follow...

Any number which is a multiple of 3, other than 3 itself, can be made up with combinations of 6's and 9's.

If the number is 44 or bigger then it must be one of the following:

a) a multiple of 3 - use 6's and 9's as above

b) one greater than a multiple of 3 - use two boxes of 20 nuggets, which leaves what's left as a multiple of 3, solve as above

c) two greater than a multiple of 3 - use one box of 20 nuggets, leaving what's left as a multiple of 3. Solve as above.

But for 43, this method fails, because when you take away the two boxes of 20 nuggets, you get 3 left over, which can't be satisfied by any combination of 6's and 9's.

That took me a few minutes to work out. Good puzzle, Icy.

I'm glad you enjoyed it, Gnomon.

Would you believe that McNuggets are at the epicentre of this particular branch of mathematical number theory?

http://mathworld.wolfram.com/McNuggetNumber.html

Maybe I'll write a guide entry about it one day.

Icy

Bravo you guys! A lovely puzzle that I was happy to leave to Gnomon.

I should say Bravi, but that would be hypernickety.

I figured that one out because 6 and 9 are both multiples of 3. I wonder what the situation would be if the three numbers were relatively prime?

I'd need to read that Mathematica page.

It's not unlike the Chinese Remainder Theorem, which lets you solve systems of linear equations in different bases.

... Just read Mathematica:

For two numbers, g(a1,a2) = a1 a2 - (a1 + a2)

For three numbers,there is no closed-form solution, but a semi-explicit solution. However, I'd need to delve into mathematical journals for it.

I was just supporting Bel.

On the other hand, if you are able to direct me to a reasonably local branch of McDonalds where I can realistically expect to be served (2 to the power of 43,112,609) -1 McNuggets, I'll agree to eat one.

I thought it didn't exist either, and I was half way to proving it when I realised I was wrong, and figured out how to work out the answer.

What does g stand for there, Icy?

Good question - Mathematica doesn't say in so many words. It appears to be the McNugget number for two coprime variables.

I'd say, g(x,y) = the largest amount of mcnuggets you cannot buy if they are sold in packs of x and of y only.

And does it mean the same in 'G spot'?

I was just going through divisors of N=a20+b6+c9 cos I didn't know what else to do when I saw the answer. Cor I thought there was going to be a line of answers.

Still, when I got over 3(a2+b3)=3K for any K belongs to natural numbers I sort of saw the logic but still was a little peeved I didn't see it.

Not too convinced this is useful knowledge though.

K>1,a,b belong to N I'm clearing out cos my brain don't work.

Useful knowledge indeed, to see that there is an elegant solution to a puzzle.