Suki Tekverk, a summer intern, spent the last two weeks here studying APL. She will be a high school senior next September and is adept in math, so in addition to APL we also considered some math problems, proofs, and proof techniques, including the following:

Given line segments `x`

and `y`

, construct (using compass and straight-edge) line segments for the following values:

` x+y`

` x-y`

` x×y`

` x÷y`

The first two are immediate. Constructing the last two are straightforward if you are also given `1`

(or some other reference length from which to construct `1`

). Can you construct `x×y`

and `x÷y`

*without* `1`

?

Constructing `x÷y`

is impossible if you are not given `1`

: From `x`

and `y`

alone you can not determine how they compare to `1`

. If you *can* construct `x÷y`

, then you can construct `x÷x`

and therefore relate `x`

and `y`

to `1`

, contradicting the previous statement.

Can you construct `x×y`

without `1`

? I got stuck (and lazy) and posed the question to the J Chat Forum, and received a solution from Raul Miller in less than half an hour. Miller’s proof recast in terms similar to that for `x÷y`

is as follows:

From `x`

and `y`

alone you can not determine how they compare to `1`

. If you *can* construct `x×y`

, then you can construct `x×x`

, whence:

` `

if `x<x×x`

then `x>1`

` `

if `x=x×x`

then `x=1`

` `

if `x>x×x`

then `x<1`

contradicting the previous statement.

I last thought about this problem in my first year of college many decades ago. At the time Norman M. (a classmate) argued that there must be a `1`

and then did the construction for `x÷y`

using `1`

. I recall he said “there must be a 1” in the sense of “`1`

has to exist if `x`

and `y`

exist” rather than that “you have to use a `1`

in the construction” or “you can not construct `x÷y`

without using `1`

”. I don’t remember what we did with `x×y`

; before Miller’s message I had some doubt that perhaps we *were* able to construct `x×y`

without using `1`

all those years ago. (Norman went on to get a Ph.D. at MIT and other great things.)