Mathematical Magical Emoji Cards

This is a mathematical magic trick using 11 cards that have an emoji on one side and a cryptic numbered circle of emoji on the back. The effect begins with all 11 cards spread on the table with the emoji-side faced down. A spectator (volunteer, relative, or student) chooses one of the 11 cards. The remaining 10 cards will themselves figure out which card the spectator had removed by executing a simple algorithm. The trick is automatic and always works no matter how often the cards are shuffled thanks to fundamental mathematical properties of addition and modulo arithmetic.

1. Get 2 sheets of A4 or letter sized card stock. White or a light color is preferable.
2. An inkjet printer - this is optional. The cards can easily be made by hand
3. If not using a printer, you will need
   1. Marker pens of various colours
   2. Circle template - this can be a bottle cap or a circular hole cut in a card.
4. A small token - this can be a small disc of paper or card

1. Using a ruler and pencil, divide the two sheets of card into a number of equal
   sized rectangles - 6 rectangles from each A4 (letter sized) sheet. The specific size is not important. The aim is to make a stack of cards that are easy to manipulate by hand - so not too big and not too small.
2. Cut out the cards. You should end up with 12 cards. You only need 11.

1. Lightly trace out a circle using a pencil. Although the circle does not have to be perfect and can be drawn by hand, you may find it helpful to use a pair of compasses, a circle template, or a bottle cap.
2. Divide this circle into 11 arcs so that you can draw 11 smaller circle equally spaced along the perimeter of the bigger circle. Position the 11 circles so that they form a left-right symmetrical pattern - that is, have one circle in the 12-o'clock position and then evenly space 5 circles on either side of that. Make sure you leave space outside of this ring of circles for the 11 emoji.
3. Choose 11 emoji or other easily identifiable symbols. You could use letters of the alphabet, pictures of your family, etc. Place these equally spaced in a ring outside of the ring of circles. It should be clear which of the 11 emoji is closest to each of the 11 circles.
4. Repeat for all 11 cards. Use the same set of 11 emoji for each card. Make sure that each emoji is in the same position along the outer ring on every card.

You may find it easier to use a printer to ensure all 11 cards are consistent. Print the emoji and ring of circles on paper and glue this to the back of each card.

1. On each card, write the numbers in the 11 circles starting from 0 and counting up to 10 in a counter-clockwise direction.
2. Important - make sure you start numbering from 0 in a different circle position for each card. No two cards should share the same numbering for each of the 11 circle positions.

The other side of each card will have one of the 11 emoji from the ring of circles you created in the previous steps. You can determine which emoji that is by the following:

1. Find the circle labelled with "0"
2. Imagine that there is a vertical mirror down the middle of the card and ring of circles. Find the circle that corresponds to the reflection of the "0" circle.
3. The emoji nearest this reflected circle is what you draw on the other side of the card.
4. Note that if the "0" is in the circle that is in the 12 o'clock position, then that circle is its own reflection. The emoji nearest the "0" circle is what you draw on the other side.

1. Show to a spectator (volunteer, relative, or student) the eleven cards, the 11 emoji on one side and the ring of circles/emoji on the other side.
2. Place all the 11 cards with emoji-side facing down. You can shuffle them just before doing this, or mix them up while on the table
3. Invite the spectator to choose one card from the eleven. They can look at which emoji they chose, but they should keep their card on the table emoji-side facing down.
4. Suggested patter: Point out what the spectator may have already noticed, that the numbered circles are all different so it won't be very impressive to identify the chosen emoji from this. What might be more impressive is that the cards themselves can figure out which emoji was chosen.
5. Ask the spectator to turn over any 5 of the remaining 10 cards.
6. Place those 5 (face-up) cards in a separate pile. Place the 5 remaining (face-down) cards in their own pile.
7. Suggested patter: The pile of 5 face-up cards is "the data". The pile of 5 face-down cards is "the code" or list of instructions. The cards will perform "an algorithm" by executing the code to process the data.
8. Invite the spectator to shuffle and/or to cut the cards in each of the piles.

1. Place a token on top of the spectator's chosen card, marking the circle in the 12 o'clock position. Suggested patter: Computers that execute algorithms have memory for storing intermediate or partial results. The spectator can use this token to keep track of intermediate results.
2. Look at the emoji on top of the data pile. Using the card on top of the code pile, find the numbered circle that corresponds to that same emoji. This number will be how many steps to move the token.
3. Move the token that same number of positions around the ring in the clockwise direction.
4. Discard the top cards of the code and data piles
5. Repeat the above 3 steps until you've discards all 10 cards.

When all 5 cards on the code pile have been "executed" against all 5 cards on the data pile, note the final position of the token on top of the spectator's chosen card.

The emoji nearest the token should be what is on the other side.

How does this trick work? Each emoji has a value such that totaling all 11 cards will give you an invariant sum. In executing the algorithm you are simply summing up the values of the 10 remaining cards. Moving the token around the ring is equivalent to subtracting the total of the 10 cards from the invariant sum, thus revealing which card was missing.

The card trick makes use of 3 mathematical properties:

1. Addition is commutative: A + B = B + A
   So it doesn't matter which card in each pair is face up or face down when you look up the number of steps to move the token. You will always get the same number.
2. Addition is associative, so that for any two pairs of numbers, this holds: (A+B) + (C+D) = (A+D) + (C+B)
   You can freely swap cards between two different pairs of cards. Even though this changes the number of steps the token moves in each pair, after applying both pairs the token will have been moved the same number of steps altogether. This is why the spectator can freely shuffle the cards without affecting the outcome
3. The values of the emoji are unique in modulo 11 arithmetic - that is, the remainder when divided by 11 is different for each emoji. Thus the token does not need to keep a running total which can be any value from 0 to 55. Instead we only need to keep track of the remainder when the running total is divided by eleven. Thus the token can simply be moved around a ring with 11 unique positions to arrive at the same result.

Acknowledgements

Thanks to James Grime for the inspiration and suggested improvements to an earlier version of this card trick.