Math Delights

From www.geom.uiuc.edu/~addingto/number_bracelets/number_bracelets.html

Unlike worksheets, this problem gets kids to practice addition and discover patterns.

Imagine that you have lots of beads with numbers 0 to 9:

Start with any two beads, e.g., 1 & 3 or 6 & 6 or or .

To determine the next bead, add the value of two beads and record only the unit digit of their sum. For example, after 1 & 3 is 4, after 8 & 9 is 7 (since their sum is 17 and 7 is in units place of the sum). Repeat this procedure, using the value of the last two beads to create the next bead in your bracelet.

A sequence becomes a bracelet when the ending two beads are the same as the starting two beads and in the same order.

Record some sequences on graph paper (with large squares), preferably with 10 or 15 columns of squares. Investigate and write up your findings.

We've been delighted how engaging children found this problem. They discovered:

• many long bracelets are the same length.
• there is a pattern of where the zeros appear.
• two odd numbers are followed by an even number.

If children get stuck, give them time to think; if they are still having difficulty making progress, consider asking some of the following questions:

1. How many different possible pairs of numbers can you use to start?
2. What's the shortest bracelet you have found?
3. What's the longest bracelet you have found?
4. What odd- and even-number patterns are there in all your bracelets?
5. If you start with two beads with different numbers, but put the beads in the opposite order, e.g., 2 4 and 4 2, do you get the same bracelets, i.e., with the same sequence of beads?
6. How many different bracelets are there?

Find comments and variations of the Number Bracelet problem on The Math Less Traveled blog (mathlesstraveled.com/?p=513), e.g., “It is easy to generalize number bracelets to moduli other than 10.at each step, add the two previous numbers and take the remainder of dividing by something other than 10.” For another version of this problem, visit www.galileo.org/math/puzzles/IrritatingThings.htm.