Four Twos

What values can you make?

Use all four twos, and the operations available from the buttons below, to construct various
mathematical expressions. Click a button to add an operator tile, and then build up expressions by
dragging the tiles on top of each other in the shaded area —
the borders will turn red when combination is possible. (You can also drag two numbers together to
join them.)

- Can you construct expressions that evaluate to each number from 1 to 10? (Don't try too hard to get 7 — it's impossible!)
- What other interesting values can you make?
- How big do you think the largest possible value might be? See if you can make it.

Paul Dirac, famous for his fundamental contributions to the development of Quantum Mechanics, developed a formula that, if logarithms are allowed, enables any positive integer result. Dirac actually did it with three 2’s, but you can always increase the number of 2’s by one with the substitution 2=√ 2×2 .

To understand Dirac's approach, note that
√√√2 = 2^{⅛} = 2^{½3} = 2^{2-3}.
Dirac used the fact that √√…√√2,
where there are n
√ signs, is 2^{2-n}, and reasoned as follows:

- Write √√…√√2 = 2
^{2-n} - Taking log base 2 gives log
_{2}(2^{2-n}) = 2^{-n} - Taking log base 2 once more gives log
_{2}2^{-n}= -n - So n = – log
_{2}log_{2}(√√…√√2 ) where there are n √ signs.