The Sign Game

The problem

Write the numbers $1$ to $10$ in a row, leaving a box between each pair of neighbours:

$1 ;\square; 2 ;\square; 3 ;\square; 4 ;\square; 5 ;\square; 6 ;\square; 7 ;\square; 8 ;\square; 9 ;\square; 10$

Into each box you must put either a plus sign or a minus sign, and then you work out the expression.

Can you choose the signs so that the expression comes out to exactly $0$? What if the row runs from $1$ to $11$? From $1$ to $12$? For which numbers $n$ can $1 \pm 2 \pm 3 \pm \dots \pm n$ be made equal to $0$?

Before you begin

Play with $1$ to $4$ first, then $1$ to $5$, then $1$ to $6$, and keep a record of which lengths work and which don't. A pattern will show up before you understand why, and that is fine: the pattern is the question.

1. With every sign a plus, the total for $1$ to $10$ is $55$. Now change the plus in front of some number $k$ into a minus. By how much does the total change?
2. So what do all the totals you can possibly reach have in common? What does that say about $0$?
3. When $0$ is not ruled out, actually find it. Look for a pattern of signs that makes any four consecutive numbers cancel, and see how far that one idea carries you.

Dissect the task

1. This problem is still elementary: no theorem is needed. Yet it sits one tier above the tiling. What is the extra conceptual reach it asks for?
2. There are two halves here: showing that some lengths are impossible, and showing that the others are possible. They call for different kinds of thinking. Which children shine at which half?
3. The habit of hunting for a quantity that never changes is called looking for an invariant. Where else in school mathematics could a child practise this habit?