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### Birthday Calendar Puzzle

```
Date: 08/29/2001 at 11:18:01
From: Nancy Kaplan
Subject: Finding the pattern.

Dear Dr. Math,

My question involves a game that I have played with my students for a
long time, yet I am always unable to explain to them why the pattern
works. I call it the birthday game. It involves putting the numbers
1-31 on a set of five different cards as follows:

Card 1
1   3   5   7
9  11  13  15
17  19  21  23
25  27  29  31

Card 2
2   3   6   7
10  11  14  15
18  19  22  23
26  27  30  31

Card 3
4   5   6   7
12  13  14  15
20  21  22  23
28  29  30  31

Card 4
8   9  10  11
12  13  14  15
24  25  26  27
28  29  30  31

Card 5
16  17  18  19
20  21  22  23
24  25  26  27
28  29  30  31

I ask the students to think of the day of the month that their
birthdays fall on, and to keep it a secret. Then I show them the cards
one by one and ask them to tell me all of the cards that contain the
number they are thinking of. Once a student has identified all of the
cards containing the number, I simply add up the top left corners of
each card and "tada" I can tell them the number they were thinking of.

Here is my question. I know that there is a pattern to explain why
this works, but for the life of me I can not figure it out. And,
obviously it works with the numbers 1-31, but is there another set of
numbers that it would work with?

Thank you in advance for your help, I look forward to hearing from
you.

Nancy Kaplan
```

```
Date: 08/29/2001 at 12:22:16
From: Doctor Peterson
Subject: Re: Finding the pattern.

Hi, Nancy.

What you are really doing is converting the birthday to binary (base
two) and back!

The card starting with 1 contains all the numbers whose 1's digit in
binary is 1:

1 = 00001 (base 2)
3 = 00011
5 = 00101
7 = 00111

and so on. (These are, in fact, all the odd numbers.)

The next card, starting with 2, contains all the number whose 2's
digit in binary is 1:

2 = 00010 (base 2)
3 = 00011
6 = 00110
7 = 00111

and so on.

Each number appears on one card for each digit that is equal to one.
If you think about what base 2 means, you can see that, for example,

6 = 110 (base 2) = 1 * 2^2 + 1 * 2^1 + 0 * 2^0
= 1 * 4   + 1 * 2   + 0 * 1
= 4 + 2

Thus, expanding the number in binary adds the numbers of the cards it
is on. In selecting the cards, the student is converting the number to
binary.

What makes this work well with the calendar is that 31, the largest
number of days in a month, happens to be one less than 2^5 = 32, so
that a nonzero 5-digit binary number can go from 1 to 31.

You could add one more card and handle numbers up to 63. See if you
can do it. (You'd also have to double the size of each of the existing
5 cards, adding more numbers to them.)

For more about Number Bases, see the Dr. Math FAQ:

http://mathforum.org/dr.math/faq/faq.bases.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
High School Puzzles

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