Bracelet Bisection

How it works

Imagine you have a bracelet like the one pictured above. The bracelet has 6 rubies, which are worth 1, and 6 sapphires, which are worth 2. If you cut the bracelet in half as shown above, one half is worth 10 and the other half is worth 8, which is a little unfair. Can you cut the bracelet in half so each half has 6 jewels and both halves are worth the same amount? Alternatively, what is the most unfair way to cut this bracelet in half?

In this activity, students explore a variety of different bracelets. For each bracelet, they first try to figure out how to cut the bracelet in half as fairly as possible, and then they try to figure out how to cut the bracelet in half as unfairly as possible. To start, the bracelets only have rubies (worth 1) and sapphires (worth 2), but students also explore bracelets with emeralds (worth 3). Students explore whether it's always possible to cut the bracelet in half perfectly fairly (so both halves have the same value), and they also try to design bracelets that it's impossible to cut unfairly — that is, no matter how you cut the bracelet in half, the two halves will be worth the same amount.

Cutting Puzzles handout

Why we like this activity

  • It’s fun! Students enjoy trying to figure out how to cut the bracelets and designing their own bracelets.

  • It helps students to develop spatial reasoning.

  • It helps students to develop numerical reasoning.

  • It requires students to engage in mathematical habits of mind:

    • Finding and using strategies to cut the bracelets as fairly and as unfairly as possible
    • Using logic to determine when it's not possible to cut a bracelet perfectly fairly
    • Finding and using strategies to design bracelets that can't be cut unfairly

  • It has a low floor and a high ceiling: Students can start making cuts by trial and error, but more challenging puzzles require more careful strategizing!