Four twos can make anything
I am currently reading the biography of Paul Direc.
In it there occurs a curious little mathematical puzzle:
The challenge is to express any who number using the number 2 precisely four times,
and using only well-known mathematical symbols.
1 = ( 2 + 2 ) / ( 2 + 2 )
2 = 2 / 2 + 2 / 2
3 = (2 + 2 + 2) / 2 = 2 × 2 - 2/2
4 = 2 + 2 + 2 - 2
According to the book, Dirac neutralised the game by producing a general formula,
which the book gives as:
n = - log2 [ log2 (2√(√ … √2))]
where the dots indicate the taking of of
n square roots.
This is rather sly (but clever), but I don’t see the logic of including the last of the twos,
without requiring a similar digit on each of the other square roots.
What do others think?
What might be the non-Dirac solutions for higher numbers?