How To Solve This Fun Math Problem?

I recently came across this fun math Olympiad problem and enjoyed solving it. When I was a school-going child, math was supposed to be something very serious.

My teachers taught the subject with a serious face while I did my best just to survive the math hour. These days, I do what I can to make math as fun as possible, as I believe both math and we deserve it.

I feel that problems like these tip the needle a little bit, which is why I like writing about them. This problem involves the following question/equation:

2¹⁴ – 1 = ?

All you need to do is compute the outcome. Do you think you can solve this problem quickly?

Spoiler Alert

I will be discussing the solution to this problem explicitly beyond this section. So, if you wish to try it out on your own first, I suggest that you pause reading this essay at this point and go ahead.

Once you are done, you may continue reading and comparing our respective approaches.

---

Powers of 2

There are a few math fundamentals that make our lives significantly easier if we know them by heart. Take multiplication tables for instance. Once we memorise the fundamental multiplication computations, we can use them to solve more complex problems quickly.

Similarly, a long time ago, when I started learning discrete mathematics, I memorised powers of 2 from 0 to 12. Ever since then, this has helped me on numerous occasions. So, I would wholeheartedly recommend that you memorise these too.

The reason why I bring this up is that this is very useful in solving this problem.

The Solution to the Problem Using Powers of 2

Let us say that you have the powers of 2 from 0 to 12 in memory. You would tell straight away that 2¹² = 4096. 2¹³ would be 4096 multiplied by 2, which would be 8192. And consequently, 2¹⁴ would be 8192 multiplied by 2, which would be 16384.

2¹⁴ = 16384

2¹⁴ – 1 = 16383

There you go! That is the answer to the problem. Time to wrap this essay up, right?

Well, there are still a couple of points that need addressing. Firstly, not everyone has the powers of 2 up to 12 in memory. Many popular discrete mathematics textbooks recommend memorising powers of 2 just up to 10.

This essay is supported by Generatebg

What then? Let us go back to my original hint to solve this problem: use algebra.

The Solution to the Problem Using Algebra

To be fair, we will still be using powers of 2 even when we employ algebra. But this approach does not require you to have higher powers of 2 in memory.

Let us start with the left-hand side of the original equation:

2¹⁴ – 1

We could write this as follows:

→ 2^(7 * 2) – 1²

This expression is of the form (a² – b²). Given this situation, we could just apply the identity (a² – b²) = (a + b)*(a – b) as follows:

→ (2⁷ + 1)*(2⁷ – 1)

We know that 2⁷ = 128 (from memory). Plugging this value into the above expression, we get:

→ (128 + 1)*(128 – 1)

→ (129*127)

→ (130 – 1)*(130 – 3)

→ (130*130) – (130*3) – (130*1) + (1*3)

→ 16900 – 390 – 130 +3

→ 16383

There you go. Note that this computation was easy for me because I knew that 13*13 = 169 from memory. This is another trick I would recommend: memorise squares of numbers from 1 to 20. It makes more complex computations easier.

That said, I hope you enjoyed solving this problem as much as I did!

---

If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you:

• How To Debunk Fake Math Proofs?
• Can You Really Solve This Tricky Logic Puzzle (II)?
• The Fascinating Story Of The Three-Body Problem

If you would like to support me as an author, consider contributing on Patreon.