You are given an elliptical-shaped land and you are asked to choose n arbitrary points on its boundary. Then you connect each point with every other point using straight lines, forming n(n - 1)/2 connections. What is the maximum number of pieces of land you will get by choosing the points on the boundary carefully?
Gustavo knows how to count, but he is just now learning how to write numbers. He has already learned the digits 1, 2, 3, and 4. But he does not yet realize that 4 is different than 1, so he thinks that 4 is just another way to write 1. He is having fun with a little game he created: he makes numbers with the four digits that he knows and sums their values. For example:
132 = 1 + 3 + 2 = 6
112314 = 1 + 1 + 2 + 3 + 1 + 1 = 9 (remember that Gustavo thinks that 4 = 1)
Gustavo now wants to know how many such numbers he can create whose sum is a number n. For n = 2, he can make 5 numbers: 11, 14, 41, 44, and 2. (He knows how to count up beyond five, just not how to write it.) However, he can't figure out this sum for n greater than 2, and asks for your help.