Become a Magician Using Powers of 2

Use your mathematical prowness to impress your friends

Introduction

I am blessed to get an opportunity to teach vedic math classes in my neighborhood for Vedic Vidyalay non-profit organization. So during one of my math class, my middle school students were requesting me to challenge them with new facts and concepts in Mathematics in general.

When I grew up, around the same middle schooler age time period there was an article in “The Hindu” newspaper about Sir Sri Ramanujam, one of the great mathematician of India. The article made such a strong impression on me that I fell in instant love with Math and even now I love solving Math problems and reading about Math.

My students’ eagerness, rekindled my ever green memories of Math and how I got impressed with many mathematical properties. One such property that helped me to show off my math skills during my childhood was the power of 2.

During the childhood, I created my very own Magic trick - “Guess what number is in my mind” from a given series of numbers.

Below is the number game that I played during my childhood and even now with my students and my son. Hope you to enjoy it and admire the beauty of numbers.

Power of 2

2 is the only number that helps you create all numbers from 1 to (2ⁿ⁺¹-1) by mere various combinations of additions of its powers.

Lets take an example to understand this beautiful property.

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

and so on…

Lets assume that n = 2, then we will take all powers of 2 from n = 0 upto n = 2.

That is… 1, 2 & 4. The total maximum value that you can form is 1 + 2 + 4 = 7.

7 is nothing but 2³-1 = 8 - 1 = 7.

So we can conclude that if you want to form numbers from 1 to 7, you will need only 3 unique values of 2 which is 1, 2 & 4. Using addition, you will be able to form all number from 1 to 7.

The below table illustrates this. If you add the column values for each row, you will get the natural number corresponding to that row.

So in general, if you want to form natural numbers series from 1 onwards, all you need is the unqiue values of powers of 2 from 2⁰ to 2ⁿ to get a maximum possible value of 2⁽ⁿ⁺¹⁾-1

Create your own Magic Game

This above property of 2 is what we are going to exploit to create our very own Magic game - “Guess the number between 1 and (2ⁿ⁺¹-1)”.

So, for the above series of natural numbers from 1 to 7, we will need 3 cards because we are taking 3 unique values of powers of 2 (1, 2 & 4).

Each card is going to hold list of natural numbers that can be created only if you add the value of these powers of 2.

What will each card contain?

1. Each card’s starting number will be a value of 2ⁿ. So the first card will have a value of 1, second card will have a value of 2 and the third one will have a value of 4.
2. Rest of the natural numbers that needs to be filled in each card will depend on the above table.
3. For every card, based on the above table, take all natural numbers that have a X mark for the column pertaining to the first value in the card.
4. For example, Card 1 will contain 1,3,5,7 as marked in the above table.

Once you fill these 3 cards, it will look like the picture below.

How to play?

Once you have the cards, you ask your friends to think any number between 1 to 7.

Once they have a number in mind, you ask your friends to say yes for all cards that has their number.

Whichever card they say YES, you simply add the first number of those cards in your mind. The final sum is the answer.

Hurray!

Lets see an example. Lets assume, I have 5 in my mind and you ask me to say yes or no for each card.

I would have said yes for Card 1 & Card 3. So all you need to do is to add the first numbers of these two cards.

1 + 4 = 5 (The number that I had in my mind)

Guess any number between 1 to 63

Here, I have already created 6 cards that can be printed and played to guess any number between 1 to 63. Enjoy.

Conclusion

My Students and my son had loads of fun with this trick and they were saying their fellow students were in awe with their magic skills.

To ensure people don’t find out your trick, you can increase the total playable number to say 63 or 127 or even more. Also don’t show the cards in the same order of power. Try asking the cards in out of sequence and above all be ready to do your abra ka dabra performances to impress your friends.

Happy Math! Happy Magic!