An observation - a pyramid of hexagons contains a cubic number of tiles

Some months back I was contemplating the board game Take It Easy. And I noticed something very interesting.

First some background... In Take It Easy, each player has a hexagonal board which looks like this:

The aim of the game is to place hexagonal tiles on the board so as to make as many complete lines of a single colour as possible. Each player has their own board and a set of 27 tiles. And the game basically plays like Bingo for statisticians.

Note that 27 is a cubic number (3 to the power of 3). This is because there are 3 different numbers in each of the 3 possible directions.

While looking at the board I realised that you could stack the 27 tiles to form a pyramid of hexagonal boards. It turned out that this was a general pattern...

The simplest pyramid of hexagons has a single hexagon, and 1 = 1 x 1 x 1.

With two layers of hexagons, you have a single hexagon on the top layer, and 1 + 6 hexagons on the second layer. Lo and behold, 1 + (1 + 6) = 8 = 2 x 2 x 2.

The third layer has 1 in the top layer, 1 + 6 in the middle layer and 1 + 6 + 12 = 19 in the bottom layer. 1 + (1 + 6) + (1 + 6 + 12) = 27 = 3 x 3 x 3.

The fourth layer has:

1 + ( 1 + 6) + (1 + 6 + 12 ) + (1 + 6 + 12 + 18) = 1 + 7 + 19 + 37 = 64 = 4 x 4 x 4.

I was pretty sure it would be easy to prove this relationship algebraically. Now algebraic proofs are great for proving that something must be true. But they often aren't great for providing insight into WHY something is true. So I set myself a much more challenging goal - find a visual representation of a cube which will show why this relationship is true.

I then forgot about the problem for 3 or 4 months. A week or two back I started thinking about the problem again. And 3 evenings later I had discovered a really beautiful visual explanation of this relationship. I hope to share that with you in the next few blog posts.

Disclosure

I've included links to Take It Easy, Take It To The Limit and the digital version of Take It Easy on Amazon.com. Please note that these are affiliate links and I could earn a small commission if you make a purchase through Amazon after clicking on one of these links.

Take It Easy is a game that is quick to play and rather addictive. It's a game I often break out at the start of a games evening while we wait for everyone to arrive. In fact I own 3 copies of this game, as this accommodates up to 12 players.

While I heartily recommend the board game, I haven't tried the digital version. I've included that link for your convenience only - a digital version is often a cheap way to decide whether a particular game is your cup of tea. But I am expressly not recommending it (nor am I recommending against it... I just don't know).