Patterns and Inductive Reasoning

Sums of Odd Numbers

Suppose we find sums of consecutive odd numbers, starting with 1:

Do you notice anything interesting about the results?

Each is a perfect square. In fact, each is the square of the number of terms being added. For example, 36 results from adding the first 6 odd numbers, and 36 = 6².

What do you think the sum of the first 10 odd numbers will equal?

Inductive Reasoning

From the pattern we are inclined to conclude that the sum of the first n odd numbers will equal n². We call such a conclusion a "conjecture," because we cannot be 100% sure it is right since we cannot check all possible sums. But it certainly seems reasonable, and if with pick a long sum, like the sum of the first 10 odd numbers, we see it is still true:

Sum of first 10 odd numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 = 10²

Arriving at a conclusion or making a conjecture based on observed patterns is called inductive reasoning. Most knowledge of how things work in the real world comes from such observation. For example, Galileo observed that objects of different weights dropped form the same height (from the leaning tower of Pizza?) landed at the same time, provided they weren't slowed down by the air like a feather would be. From this he concluded that the acceleration of gravity is the same for all matter.

Of course, it is not enough to arrive at a conclusion based on observation. To be sure it is right, we dig deeper and try to understand why it is right. Galileo argued that the weight or size of an object should not affect its acceleration due to gravity because you could think of one object as being composed of smaller objects that are touching each other. Just because they are touching shouldn't change their acceleration.

In our example of sums of odds, why do you suppose it is true? We will demonstrate on the next page by building geometric squares.

Geometric Squares

The animation below shows us that we can think of the sum of odd numbers as the area of a square built of unit squares. For example, if we already have a square of 16 unit squares (which is a 4 by 4 square) and want to make the next larger square, we would join 5 more squares on the right and 4 more on the top--that is, 9 more squares.

This is a proof, at least if we understand the properties of squares.

The Tournament Problem

Suppose there is to be a huge tennis tournament with 60 participants. It is to be an elimination tournament where they pair up at random for each round. If the number of players before a round begins is odd, then one of them (chosen at random) sits out that round. The winners of each round, and the odd one who sat out (if there is one) pair up at random to play the next round, and so on until finally there is only one winner, the champion. What is the total number of matches to be played altogether, in all rounds of the tournament?

You could solve this problem with some effort by listing all possibilities, but with 60 players that would be a lot of work.

To save time, you could look for a pattern, starting with a lesser number of players. Suppose there are only 6 players. Then for the first round they pair up and there are 3 matches, and therefore 3 winners. Now one of these winners has to sit out the next round, and the other two play each other. That is 1 match. The winner of this match plays the person who sat out, which gives us 1 more match. So the number of matches is 3 + 1 + 1 = 5.

Now what if there are 7 players in the tournament. By similar reasoning, you find that there are 6 matches, and if there were 8 players in the tournament, you would find there are 7 matches.

By examining various cases, you would discover that the number of matches is always 1 less than the number of players, so you would conclude that a tournament that begins with 60 players will have 59 matches.

This conclusion was arrived at inductively, and it is not very satisfying.

If you give it a lot of thought, it might occur to you that this actually makes sense:

Each match has a winner and a loser. Once a player loses, he/she is out of the tournament. So count the number of losers. The only person who doesn't lose is the champion, so there must be one less loser (and therefore one less match) than players.

The point of this problem is that inductive reasoning and looking for patterns can often lead us to a correct conclusion, but to really understand (or prove) the result one has to do a lot more thinking.

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