Problem
Decide odd or even without computing
The four calculations are 2+4+6+…+100, 1+3+5+…+99, 1+2+3+…+99 and 1×1+2×2+3×3+…+99×99. The actual totals are not wanted. Only odd or even has to be said. Say whether each of the four is odd or even.
Reasoning · Grade 5-2 Parity
Your answer
Watch out Many odd terms suggest an odd sum, but an even count of them makes it even.
How to solve
Strategy Look for a Pattern — Adding hundreds of numbers is out of the question, but odd and even follow a very short repeating pattern under addition, so I can work out that pattern on tiny sums first and then apply it. The key discovery is that the even terms never change anything, so the only thing worth counting is how many odd terms a sum has. That means the useful way to re-sort each calculation is into an odd pile and an even pile, and then just count the odd pile.
1STEP 1
Find the rule on short sums
Short sums teach you to track only odd-or-even.
1+3=4, 1+3+5=9, 1+3+5+7=16, 1+3+5+7+9=25
Two odd numbers always pair up into an even number, so odd terms cancel out two at a time and only a leftover single odd term can make the total odd.
2.OA.C.3Solve An Easier Related Problem2STEP 2
Note that even terms never matter
Even terms never change it.
even + even = even, odd + even = odd
An even number is made of pairs, so throwing it in never leaves anything unpaired - it cannot change odd into even or back.
3.OA.D.9Look For A PatternThe even terms never matter, so the parity of a long sum is decided by how many odd terms it holds.
Why?
Adding an even number leaves the parity alone while adding an odd number swaps it, so only the odd terms flip anything.
🧱Even and odd behave predictablyTwice any whole number is even and adding an odd amount always swaps even with odd, so a total's parity is settled by its parts before anything is computed.
Why?
The total is its terms added one after another, so the final parity is the result of all those flips in turn.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
(1) Every term is even
(1) is all even terms, so even.
2+4+6+…+100 = 2 × (1+2+3+…+50)
Every term splits into two equal halves, so the whole pile splits into two equal halves too - that is exactly what being even means.
2.OA.C.3Look For A Pattern4STEP 4
(2) Count the odd terms
(2) has 50 odd terms, so even.
100 ÷ 2 = 50 odd terms, 50 ÷ 2 = 25 pairs
If the odd terms can be lined up two by two with none left over, every pair makes an even number and the whole sum is even.
4.OA.C.5Organize Information In More Ways5STEP 5
(3) Sort the whole run into an odd pile and an even pile
(3)'s odd pile is even too, so even.
(2+4+…+98)_even + (1+3+…+99)_even = even
Re-sorting the list into evens and odds costs nothing because addition can be done in any order, and it makes the only part that matters easy to count.
2.OA.C.3Organize Information In More Ways6STEP 6
(4) A square is odd exactly when the number is odd
(4) also has an even count of odd terms: even.
odd × odd = odd, 50 odd terms → even sum
Squaring cannot change a number from odd to even, so counting odd squares is the same easy job as counting odd numbers.
3.OA.D.9Look For A PatternAnswer
even, even, even, even
50 ÷ 2 = 25
Each answer is one of the two allowed words, and none of them required a total. The reasoning can be spot-checked on shortened versions of the same calculations, where the totals are small enough to add up: 2 + 4 + 6 = 12 is even, like (1); 1 + 3 + 5 + 7 = 16 is even because there are 4 odd terms, matching the 50-odd-terms argument for (2); 1 + 2 + 3 + 4 = 10 is even, and its odd terms 1 and 3 number two, matching (3); and 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 = 30 is even with two odd squares, matching (4). Every shortened case behaves the way the rule predicts.
Takeaway
Even numbers never change odd-or-even, so just count the odd terms: an even number of odds always adds up to an even answer.
- Find the rule on short sums
- Note that even terms never matter
- (1) Every term is even
- (2) Count the odd terms
- (3) Sort the whole run into an odd pile and an even pile
- (4) A square is odd exactly when the number is odd