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 Question from Dave: My motives are simple, I want to get revenge on my brother who, unbeknown to me, repeatedly cheated when we played on-line Scrabble! Can you help? - he has posed this question knowing that I'll struggle to get the solution. Question: He is taking 6 pills a day, if he takes the pills in a different order each day, how long will it be before he takes the pills in the same order on the same date again? I think leap years might come in to it! Will I confess to using you guys? - I'll think about that one.

Hi Dave,

Suppose he is taking 2 pills a day, let's call them $a$ and $b.$ Then he can take them in the order $ab$ or $ba.$ Hence there are 2 ways to order the pills and on day 3 he will have to repeat one of the orders.

Suppose he is taking 3 pills a day. There are 3 choices for the pill he takes on the first day. Regardless of which pill he chooses he has 2 pills left and he can order them in 2 ways. Thus he has $3 \times 2 = 6$ ways to order the 3 pills. Hence there are 6 ways to order the pills and on day 7 he will have to repeat one of the orders.

Suppose he is taking 4 pills a day. There are 4 choices for the pill he takes on the first day. Regardless of which pill he chooses he has 3 pills left and he can order them in 6 ways. Thus he has $4 \times 6 = 24$ ways to order the 3 pills. Hence there are 24 ways to order the pills and on day 25 he will have to repeat one of the orders.

What about 5 pills or 6 pills?

Penny