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Twosday!

Introduction

This year, we can celebrate two “Twosdays”, on 2/2/22 and 22/2/22 (which also happens to be a Tuesday)! A great occasion for you to explore this puzzle with your students:

Can you create different integers (whole numbers) using just 2s and different mathematical operations?

You can combine 2s to create longer numbers like 22, and you can also use any other mathematical operations. Here are a few examples:

• $3=2+\frac{2}{2}$
• $11=\frac{22}{2}$
• $100=\left(2\cdot2\cdot2+2\right)^{2}$
• $5000=\frac{\left(\left(2\cdot2\cdot2+2\right)^{\left(2+2\right)}\right)}{2}$

Ideas to Consider

Depending upon the age of your students, consider some of the following:

• Create each integer using as few 2s as possible.
• Create a single integer in as many different ways as possible. Discuss with students what should count as “different.” Are these solutions for 111 different enough for your students?
• $\frac{222}{2}$ and $\frac{222}{2}+2-2$ and $\left(\frac{222}{2}\right)\cdot\left(\frac{2}{2}\right)$
• What operations do your students know? Perhaps this could be a fun time to introduce new ideas such as exponents, factorial, square roots, logarithms, and more! Here are some examples:
• $24 = (2^2)!$
• $2=(\sqrt2)^2$
• $4=\frac{\log\ \left(\left(2+2\right)^{2}\right)}{\log\ 2}$
• Challenge them to create negative integers as well.
• $-1=\frac{2}{2}-2$
• $-18 = 2-2^{2}-\left(2^{2}\right)^{2}$
• Some of the examples above used the number of 2s equal to the integer:
• 2 with two 2s: $(\sqrt2)^2$
• 3 with three 2s: $2+\frac{2}{2}$
• 4 with four 2s: $\frac{\log\ \left(\left(2+2\right)^{2}\right)}{\log\ 2}$

Challenge students to find more examples like this. Can they make 5 with five 2s? Can they make 6 with six 2s?

• For older students, it turns out that ANY integer can be created using just three 2s – using square roots and logarithms. Can you work out how?