# Croesus' Pizzas

This scrap of paper caught my eye, partially trodden in to a mush outside Bank tube station. It appears to be part of a pizza menu (illegible areas marked with square brackets):

         Extra Topping Price List               New for 2017!

[  ]   [  ]    [  ]    [  ]    [  ]    [  ]    [  ]    [  ]
12"     £1.73  £2.08   £10.65  £11.70  [  ]    [  ]    [  ]    £174
14"     £2.35  £2.86   £15.00  £16.10  [  ]    [  ]    [  ]    £238

1. Which eight toppings are advertised, and how were their prices arrived at?

2. What are the missing prices?

The toppings have inherent properties that lead to their pricing, so the answer isn't £1.73 worth of arbitrary pizza topping

Hint

I cross the road and head up Threadneedle Street as I puzzle over the fragment. Strange, it sort of looks like the price has been rounded up differently for each topping?

Hint

It's only later, back at my hotel room and packing my suitcase as neatly as possible, that the penny finally drops. Yuck!

Hint

Dough I laugh to myself. Thinking, "is that even a legal foodstuff?". I guess there's a market for everything.

• answering the question will require geometry
– jsh
Commented May 15, 2015 at 14:50
• Maybe a silly question, but are the pizza toppings actual pizza toppings? Commented May 15, 2015 at 16:52
• I've been wondering the same thing as @ShawnHolzworth. Are we looking for things like 'salami' and 'anchovies', or things which physically exist but which you wouldn't normally find on a pizza, like 'bungalow' or 'Saturn V', or even completely abstract things (such as '10%' or 'e^i' or 'love')? (Feel free not to answer if it would spoil the puzzle).
– A E
Commented May 15, 2015 at 16:55
• I'll take a Saturn V on deep crust base for £238 please.
– Bob
Commented May 15, 2015 at 17:05
• latest hint may help with that
– jsh
Commented May 15, 2015 at 17:07

The eight most commonly used coins of the United Kingdom: 1p, 2p, 5p, 10p, 20p, 50p, £1, £2. This explains your "the penny finally drops" and "dough" hints, and the "Croesus' Pizzas" title.

how were their prices arrived at?

By multiplying the value of the coin by the amount that can fit on the pizza, packing them together as closely as you can. For example, you can fit 104 2p coins on a 12" pizza*, so it costs 2.08. This explains your "packing my suitcase as neatly as possible" hint.

What are the missing prices?

According to my coin packing simulation,

         1p     2p      5p      10p     20p     50p     £1      £2
12"     £1.73  £2.08   £10.65  £11.70  £32.60  £45.50  £151    £174
14"     £2.35  £2.86   £15.00  £16.10  £42.20  £63.50  £199    £238


In 2017, the price for the £1 pie will change, because the current round coin will be replaced with a 12-sided design. This explains the "New for 2017!" header.

*I suspect we're using slightly different packing strategies, because I was able to fit 109 2p coins, not 104:

If I had to make an in-story justification for this, I'd say "the chef is clumsy and sometimes a few coins fall off during baking"

• I'll need to look at the packing discrepancy, but I'm sure your solution is correct. thanks for the in-story justification, very charitable :)
– jsh
Commented May 18, 2015 at 10:29
• I'll accept the answer but if you wanted to improve it there's a few missed hints (why did I arrive at that station and head up that street?). also some of the coins aren't circles.
– jsh
Commented May 18, 2015 at 10:29
• I guess Bank tube station and Threadneedle street are in the problem because they're near the Bank of England. I know that the 20p and 50p coins aren't circular, but devising an optimal packing strategy for beveled heptagons is a little beyond my ability, so I approximated ;-) If only they were six sided, then they would tessellate nicely... Commented May 18, 2015 at 12:05