Merchandising: How to increase your stockholding of tins (and other round items) by 15%

In the shelf of baked beans below, we fit in 48 tins when packing regularly.



However, when we resort to a 'messier' hexagonal pattern, we fit in 53 tins - that's over 10% more.



Assuming we are somewhere in the centre of the shelf, we can fit in another three tins by encroaching onto our neighbour's space on the left. Also, if this became the 'rule of the shelf', we would give our neigbour on the right, the same amount of space (the three white tins).



By working together, everyone then gets more space. On average (according to Thue's Theorem below) an increase of 14.65% (except for the one person on the end).

Mankind, in the commendable pursuit of galaxy exploration, has spent billions in the endeavour to explore the nether regions of 'space', sending a vehicle to Mars and personally stepping on the moon. However, applying Thue's Theorem we discover new reaches of space in the selling system of our local supermarket. For which shopperkind can benefit today.

Now, if only we could find a better way to pack the square items.

The Maths:
When packing tins in rows, each tin can be seen to be occupying a square (the area of which is p x r²). So, for a tin 10 cm diameter (5 cm radius) the area occupied (in a square of 100 cm²) is 22/7 x 25 = 78.5. That means 78.5% of the area is taken up by the tin and 21.5% is wasted space.
When packing tins hexagonally, the area between the tins forms a funny triangle resulting in a lot less wasted space. The formula for the area covered changes to p ÷ (2 x v3) = 22/7 ÷ (2 x 1.73) = 91% coverage (the 21.5% waste above is reduced to less than 10%).

Note: Thue's Theorem only holds for an infinite amount of space (not the case in the local supermarket) and sometimes the best array can be a mix of square and hexagonal packing (depending on the size of the tin, number of facings and dimensions of the shelf). However, hexagonal packing is always more efficient than a regular array for circular items on straight shelving.

Source: This article is based on 'Fitting circles into squares' in Rob Eastaway, Jeremy Wyndham. How long is a piece of string? Robson Books, 2003, p32.