3. Vovochka approached a slot machine, on the screen of which the number 0 was glowing. The game rules stated: «The screen shows the number of points. If you throw a 1-ruble coin, the number of points will increase by 1. If you throw a 2-ruble coin, the number of points will double. If you score 50 points, the machine will give a prize. If the number exceeds 50, all the points will be lost.» What is the minimum amount of rubles Vovochka can spend to get the prize? Answer: 11 rubles.
Solution: Let's try to solve it from the end - how to get the number 1 from 50 with the least amount of rubles, if you can only divide by 2 and subtract 1. We get: 50>25->24->12->6->3->2->1. That is, it will require 4 two-ruble and 3 one-ruble coins. It is obvious that if you use 3 two-ruble coins and fewer than 5 one-ruble coins (which corresponds to multiplying by 8), you cannot get a number greater than 40. Fewer one-ruble coins would also not suffice - this can be shown by enumeration.