Plus and minus, and parity

Anshu has been writing numbers as sums of numbers in a row (consecutive numbers): 7 = 3 + 4, 10 = 1 + 2 + 3 + 4, 12 = 3 + 4 + 5. And 15 in three different ways: 7 + 8, 4 + 5 + 6, and 1 + 2 + 3 + 4 + 5.

His questions are worth carrying with you:

• Can every natural number be written as such a sum? (Try 8, or 16, and see what happens.)
• Which numbers can be written in more than one way?
• All odd numbers are a sum of two consecutive numbers (49 = 24 + 25). What about even ones? (Two numbers in a row are always one odd and one even, so their sum is always odd.)
• Can 0 be written this way, perhaps using negative numbers?

Now a different game. Take 3, 4, 5, 6 and put a '+' or a '−' in each gap. How many expressions can you make? Eight: two choices at each of three gaps, so 2 × 2 × 2.

Work them all out. 3 + 4 − 5 + 6 = 8. 3 − 4 − 5 − 6 = −12. Do the rest. Something is the same about every answer: they're all even. (−12 is even too: a negative number with a factor of 2 is still even.)

Try four other consecutive numbers. 5, 6, 7, 8: 5 + 6 − 7 + 8 = 12, 5 − 6 − 7 − 8 = −16. All even again. In fact, take any four numbers, consecutive or not, and all eight answers share the same parity (all even, or all odd).

Explanation 1. Take one of the eight, say a + b − c − d, and flip one sign to get a − b − c − d. How much did the value change? (a + b − c − d) − (a − b − c − d) = 2b: an even amount. (So 78 + 79 + 80 + 81 and 78 + 79 + 80 − 81 differ by 2 × 81 = 162, without adding anything.)

Two numbers that differ by an even amount must have the same parity. And from any one expression you can reach the other seven by flipping one or more signs, each flip changing the value by an even amount. So all eight share a parity.

Explanation 2 is shorter:

(Same parity gives even; different gives odd.) Notice that the sign never matters: a + b and a − b always have the same parity, whatever a and b are. So a ± b ± c has one parity, and so does a ± b ± c ± d. There are endless ways to pick four numbers, and reasoning settles all of them at once, without checking a single extra case.