Open Middle®
Directions: Using the digits 1 to 9, at most one time each, place a digit in each blank 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.
There are a lot more 0% probabilities: i.e.: rolling a 1 on two 8 sided dice and rolling a 9 on two 4 sided dice. The 8-sided dice can be any number of sides that doesn’t match the numbers in the second scenario.
The 4-sided dice can be anything less than half the second roll value.