The Illusions of Mathemagics: How Tricks can be Deceptively Simple

Hello fellow math enthusiast,
My name is Robert.

If you are reading this, you hopefully have an interest, or at least general curiosity about math. If so, I'm sure you like to understand how and why the world works the way it does and why everything functions. Well, I certainly do at least.

Even the most seemingly unrelated and random things can have a simple mathematical explanation when broken down into smaller and simpler components.

That is why, for my project, I learned to perform various "magic" tricks and analyzed them through a mathematical lens in order to solve and explain how and why these tricks work.

Let's begin
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Introductory Trick

Let's start with an introductory trick to demonstrate how math can break down magic into smaller and simpler parts. Try this problem:

• Think of a number between 1-10

• Add 2

• Multiply by 2

• Subtract 2

• Divide by 2

• Subtract the original number

Did you get 1?

Now, try this with any number, not just 1-10. Still 1 right?

"Okay," you say, "but that can't always be the case, can it? There has to be an exception."

I'm here to say there are NO exceptions. Even if you try a negative number, irrational, or even imaginary number it will still be 1.

I'll show you an example, using a random negative number. Let's choose -1000.5

Crazy, right?

Now, what was the part kept constant? The 2. Here's how it works for any number:

In fact, the trick will change depending on what number is chosen to make changes to the original, but it will always be one less than that number. The following algebra proves this fact:

So, if the original number was π and it was changed with number 3 as the constant, the solution would always be 2, if it was 466/3 changed with 4 it will always be 3, or even -i⁴⁵⁰⁰ changed with 5 it will always be 4.

By throwing people off and combining numbers together, it makes the trick seem more complicated than it actually is. In reality, it is only simple algebra extended into many parts. This is Magic 101: Misdirection

For more info on this trick, click here

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Birthday Trick

Now onto a similar, slightly more complex trick. Here are the rules:

Ask someone to, secretly, write down their month of birth (1-12) and perform the following calculations:

• Double the number
• Add 5
• Multiply by 50
• Add their age
• Subtract 365
• Then, ask them the total, and secretly add 115

The first number will be the month they were born and the last will be their age (for example, January for a 10-year-old is 110 and December for a 10-year-old is 1210)

To demonstrate, let's use my birthday, which is in December. For reference, I am 16 years old at the time of writing this blog.

• m = 12 
• 2(12) = 24
• 24 + 5 = 29
• 29(50) = 1450
• 1450 + 16 = 1466
• 1466 - 365 = 1101
• 1101 + 115 = 1216

This trick works similarly to the last one, and again, is much simpler than it seems:

m = month and a = age

• m
• 2m
• 2m + 5
• 100m + 250
• 100m + 250 + a
• 100m + 250 + a - 365 = 100m - 115 + a → the total given
• 100m - 115 + a + 115 = 100m + a

(The 100 in front of the m moves the month over so age is at the end and not added onto the month.
The number will have to be multiplied by 1000 instead if the person is 100 years old or greater,
and the numbers will be shifted differently)

Essentially, the adding cancels out previous algebra, and while the number seems unrelated, they are actually being multiplied to create a certain result. To throw people off, this trick takes the "long" way by separating the algebra into steps. However, by looking at it from a different perspective, the final answer makes logical sense. At the end of the trick, it is just a number with 2 extra 0's at the end, which are filled up by the age that is simply added to the number.

All that is actually happening is shifting (usually) 1-2 numbers in the thousand's and hundred's place, and (usually) 1-2 in the ten's and one's place. But it doesn't seem that way, and that's the point of these tricks.

We are using a longer method in order for it to seem more magical.

It is much simpler to see how this trick was even initially invented once it is looked at in it's simplest terms (variables allow it to be more easily seen); while the tricks may initially seem confusing, the variables allow for the numbers to be combined and broken down into smaller portions that apply to all numbers, instead of what varies between other numbers, which makes it seem random and "mystical."

"That's great," you may be thinking, "but these are just numbers in people's heads. Is this even possible with tricks that most magicians would actually use?" As a matter of fact, it is. Allow me to demonstrate my final tricks using physical objects, specifically cards and paper.

For more info on this trick, click here

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Paper Trick

Here's the next one's rules:

• Take out three slips of paper 
• Have someone choose a starting number. Write that number on the first slip of paper.
• Write the next number on the back of that paper
• Continue the sequence of adding one to the next papers (i.e. the second paper has the next, the other side has another, and so on). There should be 6 numbers at the end, one for each side.

• Then, face up the highest number on each paper
• Add all the numbers together

For instance, if you start with 1, the sides of your cards are:

First Side - Lower Numbers

Second Side - Higher Numbers

Now, keeping them flipped on the second side (the ones with 2, 4, 6 in my case)

Let's choose one of these cards to flip back over to the smaller side. I'll choose the first one.

Option 1

These add up to 11. So what's the trick? Well, there's 2 parts. First off, any of these combinations of cards will add to 11. If you flip another one over, it's the same. Here's the other 2 options:

Option 2

Option 3

All of them are either different ways of combining 6 + 5, or however other way you want to look at it. This is because in every single case you are adding up every number and subtracting 1 exactly. Because of how addition works, it will be the same number every time from a constant difference.

However, this trick does not only work by starting with the number 1. Any other number (yes, even the annoying ones like the first problem) will work. However, it requires a little more work to get to the answer. Not every number will add up to 11; for instance, if 10 is chosen, the answers will all be 38. So, there needs to be an equation to find it.

Simple enough.

Let's represent the numbers with variables again.

These can represent any number

If all the cards remain flipped over like this, they add up to 3X + 9. By flipping one card over, you are doing the equivalent of subtracting 1 from one of these cards, as these are the higher numbers, so the equation will be 3X + 8.

This trick has a lot of variety, as you can see, but we aren't done yet. There are many other variables you can also change.

For instance, you can add and subtract another number instead of 1. If two of these cards were flipped over...

Option 1

Option 2

And so on.

In these cases, there is a constant difference of 2. Thus, the equation is 3X + 7.

You could also change the number of cards used, which would affect the first number in the equation (for instance, it would be 5X + 8 if the first trick was performed with 5 cards), or have a different amount per number (for instance, going 1, 3, 5... on the cards). These all follow the same concept, but have different results and equations that can be combined to make for an even more amazing trick!

Now for my final tricks, I will make them slightly more complicated, but not by too much. These require more effort to perform, but are easy enough concepts to grasp.

For more info on this trick, click here

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Card Trick #1

Here's card trick number 1 (with shuffled cards):

Final Results

Here's how the trick works:

• Start with a standard 52 card deck (no Jokers)
• Take 9 cards out of the deck (for instance, off the bottom)
• Pick one of the cards and place it on the top, face down, of the 9 cards
• Place those cards face down at the bottom of the deck
• Pull out 4 piles, each counting down from 10. If a card matches the number counted, leave it (for instance, in my video, I went "10, 9" and the 9 matched for the first one). If none of them match after 10 cards, put an 11th card face down on top of the pile
• Count all of the face up cards and add them together. That number will be where the chosen card will be

This trick is actually extremely simple, and only requires normal subtraction to figure out. Here's what I mean:

• 52 - 8 = 44 (it is subtracting 8 because it is the 9th card, and is included in the calculations. You are ignoring the last 8 cards in the pile to get to the top card. This trick would work the same way if the last 8 cards were removed from the deck completely.)
• The cards, at maximum, are in 11 deck piles
• Thus, it is 4(11) = the card

The numbers of the actual cards don't matter, but they are used in order to perform the trick. In this case, it works because it represents the amount of cards needed to complete a pile. Let's use mine as an example:
Pile 1: 11-2 = 9
Pile 2: 11-3 = 8
Pile 3: 11-9 = 2
Pile 4: 11-11 = 0

There are normally 11 cards in a pile, but in my first one, for example, there are only 2 cards down, meaning 9 more are needed to do the equivalent of completing the pile. Thus, to finish the pile, making the equivalent of 11 cards taken out in each, we need to take out an additional 9, 8, and 2 cards from the original deck.

¡Voilà! You get the original card back.

Performing this trick, we are actually just counting forward from the deck, instead of the 9 back like we did before, to get to the 43rd card. However, again, we are making it seem like the cards and the magicians themselves are the cause of this trick working, not simple math. The standard card deck can be manipulated and used from standard whole numbers to form a simple mathematical equation and trick.

For more info on this trick, click here

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Card Trick #2

Finally, for my last trick, I will demonstrate some more mathemagical manipulation.

Here's card trick number 2 (with shuffled cards):

Here's how the trick works:

• Start with a standard 52 card deck (no Jokers)
• Take off the top card of the deck, and count from that number to 14 (Aces are 1,  Jacks 11, Queens 12, Kings 13)
• Form 3 more piles this way
• Choose 2 of the piles (doesn't matter which ones) and put them back in the deck
• Flip one of the top cards over
• Take that number off the deck, then an additional 22 cards
• The number of cards left in the initial deck will equal the card that were not turned over

Final Results

Similar to the last one, this one only requires addition. Here's what occurred in my version:
x = top card of pile 1
y = top card of pile 2
d = number in discard pile (including the unused piles)

• The cards remaining in the piles are 14-10+1 and 14-6+1 (+ 1 because it includes that card itself as well, like a sequence), so the numbers are 5 and 9.
• So, the total cards subtracted by the amount in two piles is the total cards in the discard pile. 
• d = 52 - (5 + 9) = 38
• And, rearranging with the 22 (cards taken out) means that d = 22 + x + 6
• Substitute the d to make it 38 = 28 + x and x = 10

This one works because of the number 14. Again, the number of the card allows us to see how much of each one is remaining. Using a type of sequence, we can rearrange the piles and determine the remaining card.

For more info on this trick, click here

Even impossible and mystical tricks can be solved from learning your 1-2-3’s.

Nothing is as it seems, and even physical objects can be placed into our current understanding of math. All magic was first determined from these logical mathematical steps, and not pure coincidence. You only need to know where, and how, to look for it.

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Other tricks to look into if you enjoyed these:

The Gray Elephant from Denmark

1089 Trick