Adding odd numbers

The idea

The sum of two odd numbers is always even, while the sum of three odd numbers, or of five, is never even; the parity of a sum of odd numbers depends on how many are added.

Kishor’s puzzle

Kishor has 5 empty boxes, and each box must get exactly 1 number card. All his cards show odd numbers. The numbers in the boxes must add up to 30.

Can Kishor do it?

• Yes, with the right cards
• No, it is impossible
• We cannot tell without seeing the cards

Count the single dots

Every odd number has one single dot left over. Tap odd numbers to add them. Watch the singles pair up. Save sums with 2, 3, 4 and 5 numbers.

The singles decide

Each odd number brings one single dot. Two odd numbers: two singles make a pair, so nothing is left. Three odd numbers: three singles make one pair, and one dot is left over.

So Kishor’s 5 odd cards leave a single dot over. Their total is odd, never 30.

The sum of two odd numbers is always even, while the sum of three odd numbers, or of five, is never even. The parity of a sum of odd numbers depends on how many are added.

• How many odd numbers | Total
• 1 | odd
• 2 | even
• 3 | odd
• 4 | even
• 5 | odd
• 6 | even

Notes

The sum of two odd numbers is always even, while the sum of three odd numbers, or of five, is never even. The parity of a sum of odd numbers depends on how many are added.

Check yourself

Six odd numbers are added. Which total is possible?

Which of these sums is an odd number?

Three odd numbers are added. Each odd number leaves a single dot. Those three single dots are paired up as far as possible. How many dots are left over in the total?

Answer: 1

Three singles make 1 pair and 1 dot is left over. So the total is odd.

Meena says: “Odd numbers added together always give an odd number, because the numbers are odd.” Which sum shows that she is wrong?

• 35. 35 is odd. Six odd numbers leave six single dots, which pair up completely.
• 36 — correct. Yes! Six single dots make 3 pairs, so the total is even. For example 1 + 3 + 5 + 7 + 9 + 11 = 36.
• 41. 41 is odd. Six single dots pair up completely, so the total is even.
• 53. 53 is odd. Six single dots pair up completely, so the total is even.
• 5 + 9 + 13 + 7. Four odd numbers: four singles make 2 pairs. The total is even (34).
• 11 + 3. Two odd numbers: the two singles make a pair. The total is even (14).
• 15 + 5 + 9 + 1 + 1 + 3. Six odd numbers: six singles make 3 pairs. The total is even (34).
• 7 + 1 + 9 — correct. Yes! Three odd numbers leave three singles: one pair and one dot left over. 7 + 1 + 9 = 17.
• 3 + 5 + 7 = 15. This fits Meena’s idea (15 is odd), so it does not show she is wrong.
• 9 + 1 + 1 = 11. This fits Meena’s idea (11 is odd), so it does not show she is wrong.
• 3 + 5 = 8 — correct. Yes! Two odd numbers add up to an even number.