There is perhaps nothing which can invoke as much jealousy and bitterness as the sharing of chocolate. If you snap it into pieces it will invariably result in uneven portions and crumb spillage. Using a knife and chopping board would be fairer, if not always practical, yet this requires careful measuring, and invites similar levels of criticism as seen in the problem of cake division.
Many chocolate manufacturers have recognised this difficulty, and mould their bars into regular, easily snappable rectangular chunks. This is all well and good if the number of chunks on offer is divisible by the number of eaters. If there are leftovers, however, the problem remains.
This entry describes a little-known yet ingenious method to divide a particular shape of chocolate bar into three equal portions. The bar is the familiar triangular-shaped Toblerone. The method we will use was known thousands of years ago, in Ancient Greece.
The Toblerone bar is quite possibly the most recognisable product of Switzerland. Chocolatier cousins Theodor Tobler and Emil Baumann first produced the mountain-shaped honey and nougat-filled bar in their Berne factory in 1908. These days it comes in a variety of colours and sizes, but the shape has remained constant.
The bar is handily moulded into triangular chunks along its length; indeed, if it were not so edible it would function perfectly as a letter rack. The trouble, as previously mentioned, is that the number of chunks may not allow the bar to be divided exactly. A 50g or 75g bar, for example, has eleven chunks: being a prime number, this will only divide between either one consumer or eleven.
Through the magic of mathematical geometry, however, we will proceed to divide this bar into three equal pieces.
Divide and Conquer
You require a chopping board, a knife, a Toblerone and three hungry and bitterly competitive children.
It helps if the bar is soft, to faciliate cutting. Leaving it in a warm room for a few hours should do. You can always put the three cut pieces in the fridge afterwards to harden them, if the children are prepared to wait.
Place the Toblerone on the chopping board so you are looking at it side on. The triangular prism shape you see has 6 corners, or vertices. We will label them as follows:
- A: Left hand side, top
- B: Left, back
- C: Left, front
- D: Right hand side, top
- E: Right, back
- F: Right, front.
You need to perform the following two cuts:
A straight cut through vertices A, E and F. This will cut off one third of the Toblerone (the bit containing vertex D). One child may have this portion.
A cut through A, B and F. This will divide the remaining Toblerone in two. Give a piece to each of the unfulfilled children.
Note that all three pieces you end up with are triangular-based pyramids, but not the same shape. All, however, are the same volume. You may confirm this by weighing them if you wish.
Here's Looking at Euclid
For this method, we are indebted to the Ancient Greek mathematician Euclid. This trisection of a triangular prism into triangular-based pyramids appears as Proposition 7 in the penultimate volume of Elements, a textbook he compiled in around 300BC which documents many of the theorems of the great mathematicians of that civilisation.
To summarise Euclid's proof: as the Toblerone is a triangular-based prism with identical cross section throughout, the first pyramid we cut off, ADEF, is identical to ABCF, one of the pyramids we create with the second cut. We also know that the two pyramids we create with the second cut must have equal volume, as they divide the intact rectangular face of the prism, BCFE. Hence, all three pyramids have equal volume, QED - or as Euclid would have said: οπερ εδει δειξαι.
The beauty of Euclid's theorem is that it works with any kind of triangle. The Toblerone has a cross section which is close to an equilateral triangle, but the method works for isosceles or even scalene triangles. If the chocolatiers subsequently decide to mould their bar at a jaunty angle, we will still be able to trisect it this way.
Eagle-eyed readers may have spotted a flaw in this analysis. Because the Toblerone has gaps between the chunks, cutting it as directed will result in significantly more than three pieces, and the volume of chocolate in each trisected amount may vary slightly as a result.
This is one of the drawbacks of mathematical modelling: you have to make assumptions. We assumed that a triangular-based prism was an appropriate model for a Toblerone. Clearly it approximates to one — the box certainly is — but in the cut-throat world of chocolate division, it simply may not be accurate enough for our purpose.
Further refinement of the model may eventually produce a solution to the desired degree of accuracy, but sadly we cannot rely on Euclid's theorems to find it. Indeed, the world of solid geometry has no name for the chocolate letter-rack shape. Maybe in time it will.