The solution to the problem from the end is the algorithm for solving the problem, when the reverse calculation is performed to calculate any unknown data based on the already known final result.
A simple example
They added 1 to a certain number, multiplied the sum by 2, divided the product by 3 and subtracted from the result 4. It turned out 5. What was the number?
Decision
- 5 + 4 = 9
- 9.3 = 27
- 27: 2 = 13.5
- 13.5 - 1 = 12.5
- 9.3 = 27
Answer: 12.5
Four mathematical operations were performed in reverse order.
The same answer can be obtained by solving the equation:
More complex example.
One biologist discovered an amazing variety of amoeba. Each of them is divided into two in a minute. The biologist puts one amoeba into the test tube, and exactly one hour later the whole test tube is filled with amoeba. How long would it take for the entire test tube to fill with amoebas if at first not one amoeba was put in it, but two? [one] Solution: Since one amoeba is divided into two in a minute, and after another 59 minutes the entire tube is filled with amoeba, it takes 59 minutes for two amoeba to fill the tube.
Finding a winning strategy for games
Analysis from the end is used when searching for winning and losing situations for analyzing games . Winning is proved "from the end", using the ideas of dynamic programming : first it is proved that being in one of the "penultimate positions" you can get into the "last" (winning), then - from a certain set of "penultimate" you can only get into the "penultimate βAnd so on, until we prove that theβ penultimate ... penultimate βposition is initial. (See the Grandi function ).
Notes
- β F.F. Nagibin, E.S. Kanin . Solving problems from the end // Mathematical box. - Education, 1976. - 1 000 000 copies.