# Why is 18 = 4 + 6? Find the general rule

I have these kinds of expressions:

• Correct expressions (true and valid) \begin{align} 3 &= 3\hbox{ (it's obvious, isn't it?)}\\ 4 &= 1 + 3\\ 8 &= 5\\ 9 &= 1 + 5\\ 12 &= 1 + 3 + 5\\ 18 &= 4 + 6 \end{align}

• True but not valid expressions \begin{align} 3 &= 1 + 2\\ 10 &= 4 + 4\\ 12 &= 5 + 3 + 1\\ 14 &= 1 + 4 + 5 \end{align}

• Incorrect expressions (false and invalid) \begin{align} 10 &= 4 + 6\\ 11 &= 5 + 6 \end{align}

For each non-negative integer in the left side, there is a unique expression on the right side.

Can you tell what is the general rule? I think it is an easy one for you, guys, but feel free to ask for specific values if you need them.

• What is the difference between true and valid expressions? How can an expression be true, yet not valid? – McMagister Aug 12 '15 at 11:04
• A valid expression is written following an specific rule, and a true expression means that both sides have the same value – BianB BB Aug 12 '15 at 11:08

$3=3=F_3\\ 4=1+3=F_1+F_3\\ 8=8=F_5\\ 9=1+8=F_1+F_5\\ 12=1+3+8=F_1+F_3+F_5\\ 18=5+13=F_4+F_6$
$10=5+5=F_4+F_4$, but it should be written as $2+8=F_2+F_5$ instead. $14=1+5+8=F_1+F_4+F_5$, but it should be written as $1+13=F_1+F_6$ instead.
• @Cryol The first one is sometimes counted as $F_0$. – f'' Aug 12 '15 at 11:42