# mathematics of party favors

True story: My wife bought a bunch of party favors for a family gathering. These were rolled up gift wrapping containing some cheap present. She opened on and found a party game labeled “The Mystery Calculator”. She gave it to me.

This is what it is: A small folded cardboard mini-booklet of 6 cards of 4×8 arrays of numbers, ranging from 1 to 63.

Page 1:

$\begin{array}{rrrrrrrr} 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ 17 & 19 & 21 & 23 & 25 & 27 & 29 & 31 \\ 33 & 35 & 37 & 39 & 41 & 43 & 45 & 47 \\ 49 & 51 & 53 & 55 & 57 & 59 & 61 & 63 \end{array}$

Page 2:

$\begin{array}{rrrrrrrr} 2 & 3 & 6 & 7 & 10 & 11 & 14 & 15 \\ 18 & 19 & 22 & 23 & 26 & 27 & 30 & 31 \\ 34 & 35 & 38 & 39 & 42 & 43 & 46 & 47 \\ 50 & 51 & 54 & 55 & 58 & 59 & 62 & 63 \end{array}$

Page 3:

$\begin{array}{rrrrrrrr} 4 & 5 & 6 & 7 & 12 & 13 & 14 & 15 \\ 20 & 21 & 22 & 23 & 28 & 29 & 30 & 31 \\ 36 & 37 & 38 & 39 & 44 & 45 & 46 & 47 \\ 52 & 53 & 54 & 55 & 60 & 61 & 62 & 63 \end{array}$

Page 4:

$\begin{array}{rrrrrrrr} 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 \\ 40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 \\ 56 & 57 & 58 & 59 & 60 & 61 & 62 & 63 \end{array}$

Page 5:

$\begin{array}{rrrrrrrr} 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 \\ 48 & 49 & 50 & 51 & 52 & 53 & 54 & 55 \\ 56 & 57 & 58 & 59 & 60 & 61 & 62 & 63 \end{array}$

Page 6:

$\begin{array}{rrrrrrrr} 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 \\ 40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 \\ 48 & 49 & 50 & 51 & 52 & 53 & 54 & 55 \\ 56 & 57 & 58 & 59 & 60 & 61 & 62 & 63 \end{array}$

Here are the rules to the game.

Show the 6 cards to a friend. Ask him or her secretly pick any number from any card. Tell them to look through the 6 cards and find those cards that the number they selected also appears, and give those cards to you. You add the numbers in the upper left-hand corner of these cards. This sum will be the number your friend selected.

I haven’t proven this works but my guess is that this is based on the mathematics of the Hamming code of length 6 (see this previous post and this one).