Due 10am on Monday 6 September 2021
离散数学作业代写 It has several names and In six men’s morris, the board consists of sixteen spots ( ) arranged in two squares, as in the image below.
This Assignment is compulsory, and contributes 10% towards your final grade in Math2302, 8.5% inMath7308. It should be submitted by 10am on Monday 6 September, 2021. In the absence of a validexcuse, 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. 离散数学作业代写
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 pageof your submission. Check that your pdf file is legible and that the file size is not excessive. Files thatare poorly scanned and/or illegible may not be marked. Upload your submission using the assignment submission link in Blackboard.
1.(10 marks total) Let σ(n) denote the sum of the positive divisors of an integer Forexample,σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. In this exercise, we are interested in the function
where µ(n) denotes the M¨obius function (see page 17 of the course notes).
(a)(3 marks) Compute the values of g(n) for 1 ≤ n ≤6.
(b)(2marks) Write σ(n) as a summation Recall that means finding a function
h : Z+ → Z+ such that
(c)(5 marks) Compute the value of g(n) forany n.
2.(10marks) A class of 26 students each have a name that starts with a different letter of the English alphabet. 离散数学作业代写
The students like to sit in a long row on the riverbank to eat their lunch. Prove thatno matter how they sit down, you can find six students whose names are in alphabetical order asthey sit, read either from leftmost to rightmost, or from rightmost to (The six students do not have to be sitting right next to each other.)
3.(10 marks) The game of mills is played around the world. 离散数学作业代写
It has several names and In six men’s morris, the board consists of sixteen spots ( ) arranged in two squares, as in the image below.In this exercise we consider colourings of all sixteen spots black or white ( or). Note that this image has the rotational symmetries of a square, and the group Z4 acts on colourings of the board. Let the elements of Z4 be denoted by R0, R90, R180, R270 (where Ri denotes a counter-clockwise rotation by i degrees).
(a)(2mark) How many colourings are fixed by the identity element of D4?
(b)(2 marks) How many colourings are unchanged after rotating by ninetydegrees?
Continues overleaf. 离散数学作业代写
(c)(1 mark) Give an example of a colouring that has stabiliser made up of two
(d)(2 marks) How many colourings are unchanged after rotating by a hundred and eighty degrees?
(e)(3marks) Use the Counting Theorem to determine the number of colourings of the boardthat are distinct even after (That is, two colourings are considered distinct if one can not be rotated into the other by the action of Z4.)
4.(10marks total) Ancient Romans enjoyed games of The game Tesserae uses three six-sided dice, i.e. cubes, the faces numbered from one to six. 离散数学作业代写
The three dice are thrown at the same time, and the numbers are read off of each dice and added up. The best score is the highest sum: three sixes.
(a)(1mark) What is the probability of throwing three sixes?
(b)(5marks) Write down a generating function of the sequence sn where sn is the number of ways for the sum of the three numbers thrown to be n. 离散数学作业代写
(c)(4marks) Use a computer package (Mathematica, WolframAlpha, Sagemath, …) to com-pute which sum has the highest probability, and what this probability (Submit the text of the code you used, and what kind of package you used for the computation, as well as the numerical result.)