By Robert Freiberger from Union City, CA, USA (5dmk2-1046) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons

If a flip a coin, I get equal odds of 1/2. If I roll a die, the odds are 1/6. How can I get equal 1/3 odds?

Short answer: Roll a six-sided die. Count rolls of 4, 5, or 6 as 1, 2, or 3 respectively.

Long answer: There must be plenty of circumstances where you want equal probability split three ways. Are you having trouble deciding between pizza, burgers, or fried chicken? Are you and two friends arguing over who gets to keep the $5 you just found on the ground?

A few tricks for getting random numbers with 1/3 equal probability come to mind using coin flips or regular 6-sided die (D6). The most obvious involves relabeling the sides of the D6 so that two sides are 1, two sides are 2, and two sides are 3s. That gives you 1/3 odds for each possible result.

Another way involves flipping a coin twice. You have four possible results: heads then heads (HH), heads then tails (HT), tails then heads (TH), or tails then tails (TT). It’s simple enough to assign 1 to HH, 2 to HT, 3 to TH, and then redo the flips if you get TT.

In fact, you can use this method to get 1/6 odds with coins – it might be useful if you lose your Monopoly dice, or if everyone in your D&D group shows up thinking someone else brought the dice. Just flip the coin three times and assign six of the combinations of heads and tails to numbers and reflip on the other three. Obviously this method works best with powers of two – 1/4, 1/8, 1/16, etc won’t require reflips. It’s neat that flipping two coins can be a substitute for rolling a D4!

 

By Clément Bucco-Lechat (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

D4 showing a result of 2

If you’re eager, you can use this method to get arbitrary odds. If you want 1/x odds, just flip the coin n times such that 2n > x. This method gets cumbersome really fast though. For example, to get equal odds of 1/33 you have to do 6 flips, but 26 is 64 which means that you’ll have to reflip almost half of the time! And if assigning sequences of Hs and Ts gets confusing, you could also just resort to binary. Keep score by starting with 1 and counting a T as zero and an H as an added 2n-1, so that an H on the first flip is +1, the second is +2, the third is +4, etc. For example, TTTTT totals 1, THHTH totals 1+0+2+4+0+16=23, and HHHHH totals 1+1+2+4+8+16 =32!

 

 

Spinners or roulette wheels also work if they’re available, but I want to make this about dice. I like stuff you can throw. How do you make a D3?

We have six-sided die and four-sided die and a bunch of other shapes because there are regular polyhedrons with that many faces. Four faces is the tetrahedron, six faces is a cube, twenty is an icosohedron, etc. You can make fair dice out of any of these regular polyhedra!

By Cdang (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons

Regular polyhedra as dice

Odd numbers are hard, and generally can’t be done without curving the surface a little bit, but if you’re creative and willing to abandon planar faces you can make a die with as many sides as you’d like. Quite literally, I’m saying to think outside of the box.

Obviously there are no two or three faced regular polyhedra. The coin doesn’t count as a two sided die because it actually has three faces. While the majority of the time it is likely to land on heads or tails there is a small finite probability that it will land on the edge because a coin has nonzero thickness. Duh, it’s a cylinder! This means there is trick to make a fair three sided die out of coins. If you have one coin it’ll almost certainly land on a face, but if you have a long roll of coins it will almost certainly land on the edge. When the thickness of the stack is slightly less than the diameter of a coin a cylindrical stack of coins stuck together would have equal probability of landing on either face or the edge! We can glue some coins together and make a D3!

Lots of possibilities arise from abandoning polyhedra. For example, rather than using a triangular tetrahedron as a D4, we can use a cube by rounding out the bottom and top- it’s the principle behind a dreidel.

By Roland Scheicher (photo taken by Roland Scheicher) [Public domain], via Wikimedia Commons

Dreidel as a D4

Nothing is stopping us from making a D3 dreidel that’s based on a triangular prism with a rounded base, rather than a cube. Again, in theory, you could make a dreidel to have any number of sides! By resorting weird or rounded geometries we can make die with as many sides as we want!

 

D1, D2, and D3

D1, D2, and D3

 

There’s some simple D3s up there, even a D2 and a D1. The D1 is just a Mobius strip – why didn’t I think of that?

 

 


 

 

image credit: Wikimedia Commons

 


 

 

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