Four Application Math Problems

The following are four mathematics problems. Do the best you can on them. You are not necessarily expected to solve all these problems perfectly. I am mainly interested in how you explore problems and present your reasoning. Also, if you do not enjoy doing these problems, then you probably would not enjoy this summer program. This style of problem is one of the types of problem you would see during the program, but the subject matter is not necessarily the same.

(1) Call a number “nice” if it can be expressed as a consecutive sum of integers. For example 6 = 1 + 2 + 3 is nice. 5 = 2 + 3 is nice. 14 = 2 + 3 + 4 + 5 is also nice.

(a) Which numbers from 1 to 50 are nice?

(b) Is there a pattern? What is it for numbers beyond 50?

(c) Some numbers are “very nice” because they can be expressed “nicely” in more than one way. For example 15 = 1 + 2 + 3 + 4 + 5 and 15 = 4 + 5 + 6. Is 1000 nice? Is 1000 very nice?

(2) Call a convex pentagon with one set of parallel sides “obedient.” Call a pentagon with two sets of parallel sides “very obedient.”

(a) Can the longest side of a “very obedient” pentagon not be in one of the parallel pairs? Either way, can you prove or argue your belief?

(b) Can you come up with an alternate way to describe a “very obedient” pentagon? What else can you say about them?

(3) There are special sequences of numbers F_n that are all the fractions x/y where x and y are any numbers goes from 0 to n and y is strictly greater than x. We arrange these fractions in ascending order. For example F₄ is the sequence: 0/1, 1/4, 1/3, 1/2, 2, 3, 3/4, 1. Note some terms can be expressed multiple ways. For example 1/2 can be expressed as 1/2 or 2/4 in the above case.

(a) Compute F₅.

(b) How many different ways can a given term be expressed?

(c) Compute the difference between two successive terms of one of these sequences. So if a/b and c/d were next to each other in the sequence, what is c/d – a/b?

(d) Call two terms “split” by a later sequence when a later sequence has a term that fits between them. For example in F₄, 0/1 and 1/4 is split by F₅’s 1/5.

(4) Estimate or determine the number of possible Sudoku puzzles that can exist. If you like, start by examining “mini-sudoku” puzzles that are 2x2 blocks, such as this one:

1	2	3	4
3	4	1	2
2	1	4	3
4	3	2	1