How do you make a set of base 3 magic card?
Hello, TPHacker!
Yes, that's exactly the type of "magic cards" I'm familiar with.
. . (Use to get them in a Cracker Jack box.)
For those unfamiliar with them, here's the secret.
The base-2 cards have a simple set-up (if you know Binary).
Let's say the numbers run from 1 to 31 $\displaystyle (2^5-1)$
Write them in binary notation.
$\displaystyle \begin{array}{ccccccc}1 = 00001 & 9 = 01001 & 17 = 10001 & 25 = 11001 \\ 2 = 00010 & 10 = 01010 & 18 = 10010 & 26 = 11010 \\ 3 = 00011 & 11 = 01011 & 19 = 10011 & 27 = 11011 \\ 4 = 00100 & 12 = 01100 & 20 = 10100 & 28 = 11100\end{array}$
$\displaystyle \begin{array}{cccc}5 = 00101 & 13 = 01101 & 21 = 10101 & 29 = 11101 \\ 6 = 00110 & 14 - 01110 & 22 = 10110 & 30 = 11110 \\ 7 = 00111 & 15 = 01111 & 23 = 10111 & 31 = 11111
\\ 8 = 01000 & 16 = 10000 & 24 = 11000 & \end{array}$
On the first card, write the numbers with a "1" in the first (rightmost) position.
On the second card, write the numbers with a "1" in the second position.
On the third card, write the numbers with a "1" in the third position.
. . $\displaystyle \boxed{\begin{array}{cccc}1 & 3 & 5 & 7 \\ 9 & 11 & 13 & 15 \\ 17 & 19 & 21 & 23 \\ 25 & 27 & 29 & 31\end{array}}$ . . $\displaystyle \boxed{\begin{array}{cccc} 2 & 3 & 6 & 7 \\ 10 & 11 & 14 & 15 \\ 18 & 19 & 22 & 23 \\ 26 & 27 & 30 & 31\end{array}}$ . . $\displaystyle \boxed{\begin{array}{cccc} 4 & 5 & 6 & 7 \\ 12 & 13 & 14 & 15 \\ 20 & 21 & 22 & 23 \\ 28 & 29 & 30 & 31\end{array}}$
On the fourth card, write the numbers with a "1" in the fourth position.
On the fifth card, write the numbers with a "1" in the fifth (leftmost) position.
. . . . . . . . . $\displaystyle \boxed{\begin{array}{cccc} 8 & 9 & 10 & 11 \\ 12 & 13 & 14 & 15 \\ 24 & 25 & 26 & 27 \\ 28 & 29 & 30 & 31\end{array}}$ . . $\displaystyle \boxed{\begin{array}{cccc} 16 & 17& 18 & 19 \\ 20 & 21 & 22 & 23 \\ 24 & 25 & 26 & 27 \\ 28 & 29 & 30 & 31\end{array}}$
Effect: Have someone think of a number from 1 to 31.
. . Have him examine the five cards and give you the cards that contain his number.
. . You miraculously divine his mentally-selected number.
Secret: Add the upper-left (smallest) numbers of the cards he returns to you.
. . That is his selected number.
To make it more mysterious, scramble the numbers on each card.
Place the smallest number in another location, say, the lower-left.
And I'm sure no one will suspect any pattern or method.
Of course, this can be extended to six cards (numbers 1 to 63) or more.
Here's a set of Magic Cards (base 2) you may like to construct.
There are six number cards.
The $\displaystyle \bullet$ denotes holes to be cut out at those locations,
. . large enough to see the numbers through them.
$\displaystyle \begin{bmatrix}\\ &33 & 49 & 27 & 17 & 21 & 55 & 61 & 39 & \\ & 3 & \bullet & 31 & 51 & 63 & 43 & \bullet & 13 \\ & 15 & 7 & 1 & 19 & 15 & 23 & 59 & 41 & \\ & 57 & \bullet & 29 & 9 & \bullet & 35 & \bullet & 51 & \\ & 53 & 5 & 47 & 25 & 45 & 33 & 11 & 37 & \\ \\ \end{bmatrix}$ . $\displaystyle \begin{bmatrix}\\ &11 & 38 & 62 & 51 & 43 & 26 & 55 & 15 & \\ & 10 & \bullet & 63 & 35 & 31 & 19 & \bullet & 46 &\\& 14 & 3 & \bullet & 59 & 27 & 7 & 58 & 18 &\\& 26 & \bullet & 6 & 47 & 2 & 39 & \bullet & 22 &\\ &54 & 23 & 50 & 30 & 35 & 42 & 11 & 34&\\ \\ \end{bmatrix}$
$\displaystyle \begin{bmatrix}\\ & 5 & 47 & 28 & 53 & 61 & 13 & 20 & 52 & \\ & 37 & \bullet & 44 & 30 & 46 & 55 & 4 & 7 & \\ & 22 & 63 & \bullet & 12 & 62 & 14 & 60 & 31 & \\ & 23 & \bullet & 29 & 54 & \bullet & 15 & \bullet & 6 & \\& 45 & 36 & 39 & 21 & 47 & 28 & 63 & 38 & \\ \\ \end{bmatrix}$ . $\displaystyle \begin{bmatrix}\\ & 45 & 63 & 27& 10 & 58 & 9 & 61 & 42 & \\ & 29 & 8 & 11 & 57 & 30 & 59 & \bullet & 62 & \\ & 13 & 24 & \bullet & 60 & 40 & 47 & 14 & 56 & \\ & 45 & \bullet & 12 & 44 & \bullet & 25 & \bullet & 27 & \\ & 43 & 15 & 41 & 31 & 26 & 62 & 12 & 28 &\\ \\ \end{bmatrix}$
$\displaystyle \begin{bmatrix}\\ & 54 & 23 & 18 & 58 & 63 & 31 & 26 & 51 & \\ & 29 & \bullet & 61 & 50 & 20 & 27 & \bullet & 52 & \\ & 56 & 28 & \bullet & 17 & 59 & 48 & 21 & 60 & \\ & 31 & \bullet & 19 & 55 & \bullet & 30 & 16 & 53 & \\ & 62 & 49 & 24 & 57 & 22 & 52 & 27 & 25 & \\ \\ \end{bmatrix}$ . $\displaystyle \begin{bmatrix}\\ & 39 & 63 & 54 & 38 & 45 & 61 & 49 & 33 & \\ & 53 & \bullet & 57 & 46 & 43 & 41 & \bullet & 62 & \\ & 34 & 40 & \bullet & 55 & 42 & 51 & 59 & 35 & \\ & 60 & 32 & 44 & 59 & \bullet & 38 & \bullet & 58 & \\ & 36 & 48 & 50 & 56 & 52 & 47 & 42 & 37 & \\ \\ \end{bmatrix}$
There is a cover-card which has holes cut out at the $\displaystyle \bullet$ locations.
. . . . . . . . $\displaystyle \begin{bmatrix}\\ & .. & .. & .. & .. & .. & .. & .. & .. & \\ & .. & \bullet & .. & .. & .. & .. & \bullet & .. & \\ & .. & .. & \bullet & .. & .. & .. & .. & .. & \\ & .. & \bullet & .. & .. & \bullet & .. & \bullet & .. & \\ & .. & .. & .. & .. & .. & .. & .. & .. & \\ \\ \end{bmatrix}$
Have your volunteer select (mentally) a number from 1 to 63.
Give him the six number-cards.
Have him give you the cards on which his number appears.
When he has done so, square up the cards in your hands
. . and place the cover-card on top.
Then add the visible numbers.
Hello, t-lee!
I have never seen Magic Cards in base-3.
In fact, I've never even considered it.
I've been thinking about it since you posted your query,
. . and I think I've found them.
Let's use the numbers from 0 to 26.
Write them in base-3 notation.
.$\displaystyle \begin{array}{ccccccccc}0 = 000\\1=001\\2=002\\3=010\\4=011\\5=012\\6=020\\7=0 21\\8=022\end{array}$ . $\displaystyle \begin{array}{ccccccccc}9=100\\10=101\\11=102\\12= 110\\13=111\\14=112\\15=120\\16=121\\17=200 \end{array}$ . $\displaystyle \begin{array}{ccccccccc}18=200\\19=201\\20=202\\21 =210\\22=211\\23=212\\24=220\\25=221\\26=222\end{a rray}$
On a card, write the numbers with a first (rightmost) digit of 0:
. . 0, 3, 6, 9, 12, 15, 18, 21, 24
On a card, write the numbers with a first digit of 1:
. . 1, 4, 7, 10, 13, 16, 19, 22, 25
On a card, write the numbers with a first digit of 2:
. . 2, 5, 8, 11, 14, 17, 20, 23, 26
On a card, write the numbers with a middle digit of 0:
. . 0, 1, 2, 9, 10, 11, 18, 19, 20
On a card, write the numbers with a middle digit of 1:
. . 3, 4, 5, 12, 13, 14, 21, 22, 23
On a card, write the numbers with a middle digit of 2:
. . 6, 7, 8, 15, 16, 17, 24, 25, 26
On a card, write the numbers with a left digit of 0:
. . 0, 1, 2, 3, 4, 5, 6, 7, 8
On a card, write the numbers with a left digit of 1:
. . 9, 10, 11, 12, 13, 14, 15, 16, 17
On a card, write the numbers with a left digit of 2:
. . 18, 19, 20, 21, 22, 23, 24, 25, 26
We have nine cards.
. . $\displaystyle \begin{bmatrix} 0&3&6\\9&12&15\\18&21&24\end{bmatrix}$ . . $\displaystyle \begin{bmatrix}1&4&7\\10&13&16\\19&22&25\end{bmatr ix}$ . . $\displaystyle \begin{bmatrix}2&5& 8\\11&14&17\\20&23&26\end{bmatrix}$
. . $\displaystyle \begin{bmatrix}0&1&2\\4&10&11\\18&19&20\end{bmatri x}$ . . $\displaystyle \begin{bmatrix}3&4&5\\12&13&14\\21&22&23\end{bmatr ix}$ . . $\displaystyle \begin{bmatrix}6&7&8\\15&16&17\\24&24&25\end{bmatr ix}$
. . . $\displaystyle \begin{bmatrix}0&1&2\\3&4&5\\6&7&8\end{bmatrix}$ . . . $\displaystyle \begin{bmatrix}9&10&11\\12&13&14\\15&16&17\end{bma trix}$ . . $\displaystyle \begin{bmatrix}18&19&20\\21&22&23\\24&25&26\end{bm atrix}$
Same routine as before . . .
Have the volunteer think of a number from 0 to 27,
. . and give you the cards that contain his number.
Add the upper-left (smallest) numbers of the cards.
This can be extended to the numbers from 0 to 80 $\displaystyle (3^4 - 1)$,
. . requiring twelve cards.
Thank you so much for your help. I hate to bother you, but I have another question. You said to have someone pick a number 0-26 and then they should give you the cards that has their number on it. Once they do this you are to add the upper left numbers together? What does this do?
Say I pick the number 11 and I gave you the two cards that contained that number...are you saying I would add 2-0? If so, I don't quit understand what that means.
t-lee