Brownies...

To quash the allegations, I'm talking about the delicious baked cake-esque treat, not the club for delicious small girls. So there.

A discussion arose yesterday on the benefits of brownies - specifically the difference between "edge bits" and "middle bits".  Now, I'm a middle-bit man. I can't see the appeal of dried, crunchy, crusty edge bits of brownie when the whole point of a brownie is it's squidgy goodness. Alas, I have to concede that I'm in the minority, most people, when fighting for brownie will fight for the edge bits. Indeed, there's special tins to maximise the occurance of edges:

I can already hear the clamour of all you poor, misguided souls looking for a place to buy this small example of cooking geekery. Alas, there exists no 'edgeless' brownie tin (yet).

I am, however, lucky.

My wonderful girlfriend happens to be one of the edge-piece sheep, and so we can get along by my eating the middles bits, and her the outside, right? Surely this is an elegant solution that satisfies both parties? Wrong. We first have to devise a way of cutting up our brownie so that we each get an equal number of brownie pieces.

This leads to a veritable conundrum, how do we cut up our pan?

E = Edge, M=Middle

1x1
E  1 edge, no middle :/

2x2
E E
E E  4 edge, 0 middle

3x3:
E E E
E M E   8 edge, 1 middle (she's still fine with this, apparently)
E E E

4x4:
E E E E
E M M E
E M M E 12 edge, 4 middle (3:1, uncool)
E E E E


...I'm going to skip ahead here...


6x6:
E E E E E E
E M M M M E
E M M M M E
E M M M M E 20 edge, 16 middle (5:4, getting better...)
E M M M M E
E E E E E E


7x7:
E E E E E E E
E M M M M M E
E M M M M M E 24 edge, 25 middle (Ah-ha! we see the tables have turned..)
E M M M M M E
E M M M M M E
E M M M M M E
E E E E E E E


So, the switch happens between 6 and 7 to an edge. Of course, these are square tins. One can work out the ratio easily enough:


where n = length of a side, e = number of edge pieces and m = number of middle peices


e = 4n-4
m = (n-2)^2


And so if we declare that m = e, we can solve the above for n:


(n-2)^2 = 4n-4


<<GCSE MATHS MAGIC>>

n = 4+sqrt(8)   ~= 6.83



Now, I defy any of you to cut a pan exactly into 6.83 brownies a side. In the universal language of 4chan: Pics or GTFO.


Anyway, much more discussion was had before we had a dual revelation:


1: We don't have to have the same number on each side, and
2: Who the hell makes perfectly square baking tins anyway?!


This is game changing, and very quickly, we came to the magic number:


8x6:
E E E E E E E E
E M M M M M M E
E M M M M M M E 24 edge, 24 middle (Win!)
E M M M M M M E
E M M M M M M E
E E E E E E E E




So there you have it. If you find yourself in a situation where the ratio of edge to middle brownies is critical to the continuation of the human race, you're equipped with the tools to work it out. You can thank me later, preferably with baked goods.

EDIT

Miss Firefly_liz informs me that there's another magic ratio: 12x5.

Keep 'em coming! :)