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MATH2302/7308 Assignment 2 2022
发布时间:2022-09-01
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MATH2302/7308
Assignment 2
2022
This Assignment is compulsory, and contributes 10% towards your final grade in MATH2302 or MATH7308. It should be submitted by 2pm on Monday 5 September, 2022. In the absence of a valid excuse, assignments submitted after the due date will not be marked. (See the ECP for how to request an extension.) You should submit your assignment electronically.
Prepare your assignment as a PDF file, either by typing it or by scanning your handwritten work. Ensure that your name, student number and tutorial group number appear on the first page of your submission. Check that your PDF file is legible and that the file size is not excessive. Files that are poorly scanned and/or illegible may not be marked. Upload your submission using the assignment submission link in Blackboard.
1. (10 marks) The integers from 1 to 37 are written down, in some order, in one long row. Prove that it is possible to find seven of them that are in increasing order, or seven that are in decreasing order. (These numbers may not be right next to each other. Hint: one of the in-class exercises mentioned increasing/decreasing sequences.)
2. (10 marks total) In this exercise we consider glass window panels as in the picture below. Each of the seven segments can be made out of blue, green, or yellow glass. Our goal is to understand how many different window panels can be made.
The picture has a symmetries of an equilateral triangle. That is, the group D3 acts on colourings of window panels. They can be rotated by 0, 120 or 240 degrees (R0 , R120 , R240 respectively), or flipped along three lines (F1 , F2 , F3 ). (The three circles in the picture are centred at the points of an equilateral triangle, and their radius is the edge length.)
(a) (2 marks) How many window panels are fixed by the identity element of D3 ? (b) (2 marks) How many panels are unchanged after rotating by 120 degrees?
(c) (2 marks) Give an example of a window panel that has a stabiliser of two elements.
(d) (4 marks) Use the Counting Theorem to determine the number of different window panels that can be created. Two panels count as different if they can not be made to look alike after rotations and flips.
3. (10 marks total) The Mint of Bergengotia issues coins of 1, 2, and 4 Forints. We have three 1 Forint coins, five 2 Forint coins and seven 4 Forint coins in our pocket.
(a) (5 marks) Let cn be the number of ways we can pay n Forints using a few of our coins. Write down a generating function for the sequence (cn )n>0 . (No need to simplify this generating function. Two payments are different if they use a different number of at least one type of coin. The order of the coins does not matter.)
(b) (5 marks) Use a computer package (Mathematica, WolframAlpha, Sagemath, . . .) to find
the number(s) n so that cn is largest. (Submit the text of the code you used, and what kind of package you used for the computation, as well as the numerical result.)
4. (10 marks) Use a generating function to find a closed form for the sequence (an }n>0 defined recursively as:
a0 = 2, a1 = 3, an+2 = 6an - an+1 (n > 0).