Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to complete this sentence:

Rolling a sum of ___ on two ___-sided dice is the same probability as rolling a sum of ___ on two ___-sided dice.

### Hint

How could we organize the possible values to see this visually?

How many possible answers are there?

How many possible answers are there?

### Answer

Thank you to the secondary mathematics teachers from San Marcos Unified School District for these great answers:

Rolling a 9 on two 6-sided dice is the same probability as rolling a 2 on two 3-sided dice.

Rolling a 2 on two 4-sided dice is the same probability as rolling a 5 on two 8-sided dice.

Rolling a 3 (or 7) on two 4-sided dice is the same probability as rolling a 9 on two 8-sided dice.

Finally Robert Gordon found that you get a 0% probability from rolling a 9 on two 1-sided dice which is the same probability as rolling an 8 on two 3-sided dice.

Rolling a 9 on two 6-sided dice is the same probability as rolling a 2 on two 3-sided dice.

Rolling a 2 on two 4-sided dice is the same probability as rolling a 5 on two 8-sided dice.

Rolling a 3 (or 7) on two 4-sided dice is the same probability as rolling a 9 on two 8-sided dice.

Finally Robert Gordon found that you get a 0% probability from rolling a 9 on two 1-sided dice which is the same probability as rolling an 8 on two 3-sided dice.

Source: Audrey Mendivil, Daniel Luevanos, and Robert Kaplinsky

Is it possible to construct a 3 sided die?

Yeah, crazy enough you can make 3-sided dice and even 2-sided dice. Here’s a link to the 3-sided die: https://www.cnet.com/news/this-3d-printed-3-sided-die-is-a-work-of-modern-art/

You can also roll a sum of 2 on 3-sided dice and a sum of 5 on 6-sided dice, both of which have a probability of 1/9.

Rolling a nine on two six sided dice is the same as rolling a 2 on two 3 sided dice.

Rolling a 2 on two 4-sided dice is the same as rolling a 5 on two 8-sided dice

Rolling a 9 on two 6-sided dice is the same probability as rolling a 2 on two 3-sided dice.

rolling a 2 and a 4 and a 5 and a 8 is the same.

Rolling a 9 on two 6-sided dice is the same probability as rolling a 2 on two 3-sided dice.