Introduction to Coloring Technique Through a Magic Trick
By: Alvin Piter
Tags: Puzzle, Card Game, Mathematics, Logic, Magic

Hello, world!

My friend sent me this video yesterday. Please watch the video first and try to follow along.

This trick is really cool, kudos to Cyril for preparing it. Cyril is a great street magician, I’ve watched many of his videos on YouTube. I recommend you to watch his burger trick.

My friend saw this trick on TV a couple of days ago and she believes there is a math trick behind it. Out of curiosity, I tried to crack it. And indeed there is one, it is called the coloring technique. In this article, I will walk you through my thought process for solving this trick. You might learn a thing or two (or none ☹) along the way. Buckle up!

The trick

Let’s break down the trick into smaller actions:

The audience pick a card

The audience locate their card on the glass

Action 1 : Cyril rolls a die and asks the audience to move as many times as the number showed up by the die. Each move is either up, down, left, or right. For example, if the die shows 2, then the audience will move 2 times.

: Cyril rolls a die and asks the audience to move as many times as the number showed up by the die. Each move is either up, down, left, or right. For example, if the die shows 2, then the audience will move 2 times. Action 2 : Cyril removes all the cards which can’t be the card the audience lands on after Action 1.

: Cyril removes all the cards which can’t be the card the audience lands on after Action 1. Action 3 : Same as Action 2, but this time the audience can only move up or down.

: Same as Action 2, but this time the audience can only move up or down. Action 4 : Cyril removes all the cards which can’t be the card the audience lands on after Action 3.

: Cyril removes all the cards which can’t be the card the audience lands on after Action 3. Action 5 : Cyril asks the audience to move as many times as the number on the card the audience is currently on. Each move is either up, down, left, or right.

: Cyril asks the audience to move as many times as the number on the card the audience is currently on. Each move is either up, down, left, or right. Action 6 : Cyril removes all the cards which can’t be the card the audience lands on after Action 5.

: Cyril removes all the cards which can’t be the card the audience lands on after Action 5. Action 7 : Cyril asks the audience to move clockwise as many times as the number on the card the audience is currently on.

: Cyril asks the audience to move clockwise as many times as the number on the card the audience is currently on. Audience lands on Eight of Diamonds

Before we dive deeper…

There are a couple of things I’d like to point out:

Notice that all cards on the glass have different numbers on them, so I will refer to a card only by its number (ignoring the suit).

I will categorize the cards into 2 categories, center cards and corner cards. Center cards are the cards which are painted green, while corner cards are the cards which are painted yellow.

Green cards are called center cards, while yellow cards are called corner cards

A move is when the audience makes 1 step to the neighboring card. It may be in any direction (up, down, left, or right).

After the first 4 actions, the audience will be on one of the center cards

Cyril said it himself on minute 1:02 of the video:

And also, if you happen to land on one of these corner cards, just simply count back the way that you came.

With that statement, we can conclude that audience can’t land on any corner cards. After Action 4, the audience will be on one of the center cards. Later on, I will omit corner cards from the image.

After action 5 and 6, the audience will be located on an even-numbered card

Here comes the coloring technique. Let’s paint the center cards with either red or blue in the manner of a chessboard. Like this:

Paint the center cards red or blue in a manner of a chessboard

Observe that:

Observation 1: If the audience make a move, they will land on a card which color is different with the color of the card they were originally on. Proof: Because all neighboring cards has different color.

From the previous observation we can conclude:

Observation 2: If the audience moves odd number of times, they will land on a card which color is different with the color of the card they were originally on. Proof: It is similar to turning light on and off. If the light is initially off and you toggle it odd number of times, the light will be on in the end.

and

Observation 3: If the audience moves even number of times, they will land on a card which color is the same with the color of the card they were originally on. Proof: Similar to previous proof.

At the beginning of Action 5, the audience may be on any center cards. There are 2 cases:

At the start of Action 5, the audience is located on an odd-numbered card. All the odd-numbered cards are painted red (coincidence? Not!), which means if the audience starts from one of these odd-numbered cards they will land on one of the blue cards. This is due to Observation 2.

At the start of Action 5, the audience is located on an even-numbered card. All the even-numbered cards are painted blue, which means if the audience starts from one of these even-numbered cards they will land on one of the blue cards. This is due to Observation 3.

I put this image here to help you visualize the above 2 statements

In either case, the audience will be on one of the blue cards after Action 5. This is why in Action 6 Cyril can safely remove card 3 because it is painted red.

Action 7 is straightforward

At this point, the audience will be on one of the blue cards.

There are 4 possible cards on which the audience may be on:

Two (of Clubs). The audience moves clockwise 2 times and they will land on Eight (of Diamonds).

Four (of Clubs). The audience moves clockwise 4 times and they will land on Eight (of Diamonds).

Six (of Spades). The audience moves clockwise 6 times and they will land on Eight (of Diamonds).

Eight (of Diamonds). The audience moves clockwise 8 times and they will land on Eight (of Diamonds).

In either case, the audience always lands on Eight of Diamonds. Voilà.

My expression when I solve this trick

Final words

If you find anything incorrect in this article, please let me know. Thank you.

Alvin out. Bye.