A rooster was very happy. When he saw his master coming, Sammy gave it to him. It hopes to eat rice every day. The next day, the master gave it rice, the third day, and the ninety-ninth day. So the rooster thinks that his master will always give him rice. On the hundredth day, the rooster saw his master coming and thought there was rice to eat again, but the master caught it and killed it.

It is obviously incomplete for roosters to conclude that they have food every day. The cock's wisdom is limited, and it is impossible to reach such an understanding.

In mathematics, incomplete induction is often used to discover laws. However, as I said before, the conclusion reached in this way must be strictly proved before it can be established.

The conclusion related to natural number n is often proved by mathematical induction. Mathematical induction is also called complete induction, which can be abbreviated as induction without confusion. It is divided into two parts: first, consider the simplest case, which usually proves that the conclusion is valid when n= 1 This step is called laying the foundation stone. Second, consider whether the next step can be introduced from the previous step. That is to say, if the conclusion holds in n- 1, then the conclusion also holds in n, and this step is called induction.

If these two parts are completed, the conclusion can be established when n = 1 and when n = 2; The conclusion holds when n = 2, and it is deduced that the conclusion holds when n = 3. If we push it step by step, we can conclude that it holds true for all natural numbers.

The idea of mathematical induction has been used many times before. For example, in the third part, we actually prove that 2n(n≥4) matches can be combined in pairs according to the rules. At that time, the practice was to start from the simplest situation and combine 8 matches in pairs (n = 4). This is the cornerstone.

Then, for fourteen matches, we once attributed it to the problem of twelve matches; Twelve roots boil down to ten; Ten roots boil down to eight. This step by step dates back to August, that is, from August 1 to 14. Similarly, it is also possible to increase to 40, or more generally to 2n. The key is to combine the fourth match from the left with the first match. In this way, the problem of 2n matches becomes the problem of 2(n- 1) matches. As long as 2(n- 1) matches can be combined in pairs, 2n matches can be combined in pairs. This is the second part: induction.