Deductive Reasoning: Proving Odd + Odd Is Always Even

0:00Is the sum of two odd numbers always even?

0:04Test a few cases. 3 plus 5 is 8, 7 plus 9 is 16, and 11 plus 15 is 26.

0:14All even. But we can never test every pair of odd numbers.

0:19A proof uses deductive reasoning: it goes from facts that are always true to a conclusion that must be true.

0:28An integer is a whole number, like minus 3, 0 or 5. Even numbers are 2 times an integer.

0:37An odd number is one more than an even number, so two odd numbers are 2a plus 1 and 2b plus 1, where a and b are integers.

0:48Add them: the sum is 2a plus 2b plus 2.

0:54Take out the common factor of 2: 2 times, a plus b plus 1.

1:00Since a and b are integers, the bracket is an integer, so the sum is 2 times an integer: always even.

1:09Check with 3 plus 5: a is 1, b is 2, and 2 times 4 is 8. It matches.

1:18A common mistake is proving it with one example. 3 plus 5 is 8 checks only one case. A proof covers every pair of odd numbers.

1:29A puzzle: players take turns removing 1, 2 or 3 counters. Taking the last one wins.

1:37Reason backward from the end. Suppose you leave your opponent 4.

1:42They take 1, you take 3. They take 2, you take 2. They take 3, you take 1.

1:50Each round takes 4 in total, so leave 12, then 8, then 4.

1:5712, 8 and 4 are multiples of 4. Leave one every time, and the last counter is yours.

2:05Examples can suggest a pattern, but only a proof shows it is always true.