Slice Of Life

Floss had a post yesterday involving the number 191. I knew 191 was a prime, but wondered if there was anything else interesting about it. Which is when I discovered it was a "Lazy Caterer's Number"

I had never heard of the LCNs before, but here is how it works - imagine getting a pizza - and making 3 cuts across the centre, giving six slices like the one in this box.

Now if you were being lazy you may not cut as accurately. Your lines would not intersect at the centre, and you could end up with seven pieces [albeit oddly shaped] like this...

The so called Lazy Caterer's Sequence is the pattern of numbers representing the maximum number of pieces that can be made with a given number of straight cuts.

The mathematical formula [where p = pieces and n = cuts] is given by 

p = (n² + n + 2) ÷ 2

and the sequence is 1, 2, 4, 7, 11, 16, 22, 29, 37, 46 ... and the term for 19 cuts is 191. The boring term for this sequence is the central polygonal numbers.

I admit that I am utterly fascinated - and wonder why have I never come across this before? [I am such a maths geek that the Fibonacci series and Pascal's triangle were very significant numbers when Bob and I began courting]

The LCN appears to have such practical use - how often are you serving up a pizza/cheesecake/gateau/whatever and after you have neatly portioned it out for the guests present, one says "Oooh, no, just a sliver for me, I am dieting" whilst another says "Well, I am absolutely starving!" The picture above would indicate an ideal way to serve 3 average people, 2 dieters and 2 ravenous people!

Oh dear, this is just making me hungry for cake...