# mathematical way to determine what “group” to choose

I could fudge something together, but wondered if there was a proper way to accomplish this.

Say I have three sets of boxes, and each day I pick from a different box, and then rotate

• Box 1 Day1 Day4 Day7
• Box 2 Day2 Day5 Day8
• Box 3 Day3 Day6 Day9

What would be the best way to determine what box to use on what day?

In simple terms, I know i am on day 5, how do I know to use box 2

If there is a good solution, what "area" of maths is it (sorry, been a while!)

Thanks

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I assume that you mean for days to go on and on, so Box 1 is chosen on days 1, 4, 7, 10, 13, 16, and so on, with similar results for the other boxes. Then to find the Box number on day $n$, do this. (i) Find the remainder $r$ when $n$ is divided by $3$. (ii) If $r=1$, you use Box 1. If $r=2$, use Box 2. If $r=0$ use Box 3. The answer would be a little nicer if you renamed Box 3, called it Box 0. The subject is, sort of, elementary number theory. Most computer languages have an operation that calculates the remainder when $n$ is divided by $k$. –  André Nicolas Oct 7 '11 at 15:58

Modular arithmetic: rename Box 3 as "Box 0", and divide the day number by 3 with remainder.

Day 5: $5 = 3(1) + 2$, remainder is 2, so you use Box 2.

Day 16: $16 = 3(5) + 1$, remainder is 1, so you use Box 1.

Day 96: $96 = 3(32) + 0$, remainder is 0, so you use Box 0 (that is, Box 3).

If the day number is $n$, then there is a unique $a$, $a=0$, $1$, or $2$, with $n \equiv a\pmod{3}$. You want Box $a$.

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thanks to you and everybody else for the help! –  Keeno Oct 7 '11 at 16:13

You want the remainder on division, so on day 17 you would divide 17 by 3, getting 5 with a remainder 2 and use box 2. It is easier if you number from 0, otherwise you need to remember that instead of 0 you use 3.

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