## Odd dice

By David Shaw, 11 December 2019

**Difficulty: Taxing**

I have a collection of strange dice.

- One is a regular dice, with sides numbered: 1, 2, 3, 4, 5, 6.
- One is a doubling cube, with sides numbered: 2, 4, 8, 16, 32, 64.
- I also have two Sicherman dice, which are numbered: 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8.

If I roll all four dice and multiply the results together, what’s the probability of getting an odd number?

For an extra challenge, what’s the probability that the sum of all four is an odd number?

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

For the multiplication, look at the odd and even sides of each dice – are any of the dice special?

For the addition question, the regular dice is the key!

## Brainteaser answer

For the multiplication question, start by looking at the doubling cube. The doubling cube only has even numbers on it, so it will always roll an even number. When you multiply by an even number, you always get an even answer. So the product will never be odd!

## Extra challenge

For the addition question, let’s look at the regular dice. It has three even sides and three odd sides, so it has a 50% change of rolling even and a 50% chance of rolling odd.

If you add the other three dice together, the answer will be either even or odd. If the three add to an even number, there’s a 50% chance that the regular dice will be odd and make the sum odd. If the three add to an odd number, there’s a 50% chance the regular dice will be even and make the sum odd.

So in either case, there’s a 50% chance that the sum will be odd!

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