For @Itchymaths : I had a simply lovely "everyday maths" interaction with my kids (9 and 12) on the topic of ice cream flavours. Our starting problem; with the 3 types of ice cream on offer at the restaurant, how many "2 scoop" options are there to choose from?
(obviously, vanilla + ginger is the same as ginger + vanilla). Both kids happily did 3 types in their head (6). Made harder when we added the 3 sorbet flavours in, and so had to do 6. With a napkin they worked it out building relatively sensible triangles
So we continued about whether we could solve for "any number" of flavours of ice cream. First a long argument about whether we should use "i" for ice cream, "t" for "type of ice cream" or "curly x" because "that's what real mathematicians use" (the last point from my daughter)
We did this by drawing out the options, and looking at symmetries. I did this as "t^2 overall, take away the leading diagonal, halve, then add back the leading diagonal" - which, on rearranging, gives one t(t+1)/2 - is there an easier explanation which does not need rearranging?
We then did entirely practical things, like working out the solution to 156 different types of ice cream, and whether it was possible to have 156 different types of ice cream (after consideration, yes due to the infinitesimal changes in ginger / vanilla and other flavours).
(this is obviously the infinite ice cream flavour lemma)
We discussed generalising into a 3rd scoop - which I described as the 3rd dimension of ice cream - and this had them in hoots of laughter - and I was about to knuckle down to the general, n-dimensional ice cream problem when they posed a separate problem
This is due to the innovation in the restaurant for "half scoops", where two half scoops could be combined; two whole scoops, 4 half scoop choices, but now more complex symmetry relationships
eg (Vanilla, Ginger),(Mint,Chocolate) is the same as (Chocolate,Mint),(Vanilla,Ginger), but obviously is completely different from (Chocolate,Ginger),(Vanilla,Mint).
So - I'd like to pose the general question for you: for t types of ice cream, arranged in n scoops with p partitions per scoop, how many different deserts can the restaurant serve? (to be concrete, t=156, n=3, p=2)
