What about adding 3 odd numbers? Can the resulting sum be arranged in pairs?
Answer
No. The sum of 3 odd numbers is always odd, so it can never be arranged in pairs.
odd + odd = even (the two leftover dots join into a pair)
even + odd = odd (one dot is left over again)
So odd + odd + odd = odd
Example: 5 + 7 + 9 = 21, and 21 dots leave one dot alone.
Why it happens: the three leftover dots cannot all be matched — two of them make a pair and the third has no partner.
Q2.
Explore what happens to the sum of (a) 4 odd numbers, (b) 5 odd numbers, and (c) 6 odd numbers.
Answer
Number of odd numbers
Example
Sum
Parity
(a)
4
1 + 3 + 5 + 7
16
even
(b)
5
1 + 3 + 5 + 7 + 9
25
odd
(c)
6
1 + 3 + 5 + 7 + 9 + 11
36
even
(a) even (b) odd (c) even
Why it happens: the leftover dots cancel two at a time. With 4 or 6 odd numbers every leftover finds a partner; with 5 odd numbers one dot is always stranded.
Q3.
Two siblings, Martin and Maria, were born exactly one year apart. Today they are celebrating their birthday. Maria exclaims that the sum of their ages is 112. Is this possible? Why or why not?
Answer
No, it is not possible.
Born one year apart ⇒ their ages are consecutive numbers.
In any two consecutive numbers, one is even and the other is odd.
even + odd = odd
But 112 is even. So the sum of their ages can never be 112.