Andrei from School 205, Bucharest, Romania and Yatir from Maccabim-Reut High School, Israel both spotted the pattern in the squares (in binary) of numbers that are expressed in binary by using only `ones'. Andrei proved the rule using induction and Yatir proved it using geometric series.

First this is Andrei's description of the rule.

I start by considering the successive multiplications in base $2$:

$1 \times 1$	$=$	$1$
$11 \times 11$	$=$	$1001$
$111 \times 111$	$=$	$11001$
$1111 \times 1111$	$=$	$11100001$
$11111 \times 11111$	$=$	$1111000001$

Now I guessed the rule:

$$\overbrace {11... 11}^{n ones} \times \overbrace {11... 11}^{n ones} = \overbrace {11... 11}^{(n-1) ones}\overbrace {00... 00}^{n zeros }\overbrace{1}^{1 one}.$$

This means that squaring a number containing only '$1$s', written in base $2$ we obtain a number containing ($n$-1) digits of '$1$', followed by $n$ digits of '$0$' and a last digit '$1$'. The product has $2n$ digits.

Andrei then proved this rule by mathematical induction but first let's see how Yatir proved it. Here is Yatir's solution.

In this proof $m_n$ will mean that the number $m$ is in base $n$.

Just like every decimal number can be expressed as a sum of powers of $10$: ( $5432_{10} = 5 \times 10^3 + 4 \times 10^2 + 3 \times 10^1 + 2 \times 10^0$ ), every binary number can be expressed as a sum of powers of $2$: ($1010_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0$).

So $1111_2$ is the sum of four powers of $2$:

$1111_2 = 2^3 + 2^2 + 2^1 + 2^0 = 2^4 - 1$.

The binary number written with $n$ ones is the sum of $n$ powers of 2 (a geometric series) equal to $2^n - 1$.

So $N^2$ in binary is (from the left) a string of $n-1$ ones and after them a string $n$ zeros and after them $1$.

Now this is Andrei's rather different proof:

Now I have to prove this formula by induction. We assume the result is true for $n = k$ and then square a number with $(k+1)$ digits, all '$1$'. In base $2$:

To get the last line above I added the digits order by order. This proves the result is true for all integers by the method of induction