Making any integer with four 2s

February 22, 2025 at 14:53 Tags Math There's a cute math puzzle that can be interesting to folks on very different levels: Given exactly four instances of the digit 2 and some target natural number, use any mathematical operations to generate the target number with these 2s, using no other digits. Some examples can be done by elementary school kids: \[\begin{align*} 1&=\frac{2+2}{2+2}\\ 2&=\frac{2}{2}+\frac{2}{2}\\ 3&=2\cdot2-\frac{2}{2}\\ 4&=2+2+2-2\\ 5&=2\cdot 2 +\frac{2}{2}\\ 6&=2\cdot 2\cdot 2 - 2\\ \end{align*}\] In middle school, kids learn about exponents, factorials, etc. which expands the range considerably: \[\begin{align*} 18&=2^{2^{2}}+2\\ 28&=(2+2)!+2+2\\ 256&=(2+2)^{2+2}\\ 65536&=2^{2^{2^{2}}}\\ \end{align*}\] Then come the tricks; for example, the number 22 (twenty two) can be seen as a valid use of two 2s, and so on; so we can have: \[\begin{align*} 26&=22+2+2\\ 11&=\frac{22}{\sqrt{2+2}}\\ 444&=222\cdot 2\\ \end{align*}\] Getting to 7 is notoriously difficult, ... Read full article.