# Modular counting and the table setting problem

We began this session with a big reminder about modular thinking. In math, this means saying two numbers are the same when they really differ by a fixed amount. For example, 26 o’clock is really 2 o’clock because they differ by 24 hours. Using this reasoning again and agin, we can answer questions such as: what time of day will it be in a million hours?

Now here’s the fun investigation: suppose that N of your friends are seated around a circular table. Each place setting has a nametag, but very few of them match the person sitting there. In fact, only one matches. One friend suggests spinning the table to try to get more matches. But no matter how far you spin the table, just one nametag matches. For which N is this possible? What does it look like?

Our investigation led to a number of interesting table-setting patterns (the nametag arrangements), as well as some numerical patterns (the numbers N that worked).