University of New Hempshire The Sudoku Puzzle Quick Question

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There are two quick questions about sudoku puzzle. It looks very long, but you will understand that this is the usual style of puzzle topics if you are good at solving such problems. I need the experimental tutor to help me, and it won’t spend you 2mins for each small question if you are good at it. Attached is some samples and notes it might help you to solve questions.Just paying attention on reading question carefully and answers them as detailedly as you can. Thank you for your attention.

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Problem 3. (35 Points) (Sudoku puzzle) (see https://en.wikipedia.org/wiki/Sudoku) In this
problem, we explore the inherent connection between Sudoku puzzles and logic. Consider a 4×4
Sudoku puzzle represented by a 4 x 4 grid made up of four 2 x 2 subgrids, known as blocks. For each
puzzle, some of the 16 cells, called givens, are assigned one of the numbers 1,2,3,4, and the other
cells are blank. The puzzle is solved by assigning a number to each blank cell so that every row,
every column, and every one of the four 2 x 2 blocks contains each of the four possible numbers.
For each of the following questions, you MUST adequately explain your reasoning to receive full
credit.
To simplify notation, let
Ň
i=1
denote pi V P2 V… V Pr and
n
Api
i=1
denote pi 121…Apr. You MUST express your answers below using this condensed notation when
applicable.
a) (5 Points) To encode a Sudoku puzzle, let pli,j,n) denote the proposition that is true when the
number n is in the cell in the ith row and jth column. For example, if the number 3 is given as the
2
value in the first row and first column, then we set p(1,1, 3) to be true, and hence p(1,1, n) is false
for n 73. In a 4 x 4 Sudoku puzzle, how many propositions pli,j,n) are there for i, j, n= 1, 2, 3, 4?
b) (0 Points) We assert row i should contain the number n for any fixed i = 1, 2, 3, 4 and fixed
n=1,2,3,4. Express this information using the proposition pſi, j, n) and the appropriate logical
operator(s).
Solution. The solution for this part is given to you. For a fixed row i and fixed number n, n
could be in the cell in row i column j for j = 1,2,3, or 4.
V pli,j,n).
j=1
c) ( Points) Express the assertion that a fixed row i contains all numbers n for n = 1, 2, 3, 4 using
the proposition pſi, j,n) and the appropriate logical operator(s). (Hint: take advantage of your
solution in (b).)
Solution. The solution for this part is given to you. Can you figure out why?
A V pli,j,n).
n=1j=1
d) (5 Points) Express the assertion that every row i contains every number n for i, n = 1, 2, 3, 4
using the proposition pſi, j,n) and the appropriate logical operator(s). Explain your reasoning.
(Hint: take advantage of your solution in (c).)
e) (5 Points) Similarly as above, now express the assertion that every column j contains every
number n for j, n= 1, 2, 3, 4 using the proposition pſi, j,n) and the appropriate logical operator(s).
f) (5 Points) Express the assertion that each of the four 2 x 2 blocks contain every number n for
n= 1, 2, 3, 4 using the proposition pijn) and the appropriate logical operator(s). (Hint: draw a
4 x 4 grid and determine how to index each 2 x 2 subgrid.)
g) (5 Points) Express the assertion that if the cell in the ith row and jth column contain the
number n, then it must not contain the number n’ for n’ n using the proposition pli, j, n) and
the appropriate logical operator(s). (Hint: use implications.)
h) (5 Points) Express the assertion that no cell contains more than one number using using the
proposition pſi, j, n) and the appropriate logical operator(s). (Hint: use the result from the previous
part.)
i) (5 Points) Deduce that a solution of a given 4 x 4 Sudoku puzzle can be found by solving a
satisfiability problem. (Recall the meaning of satisfiability introduced at the end of lecture 3. You
do not need to know the rigorous definition of satisfiability to answer this question.)
2. writing neatly and easy to recognize.
3. make your answer logically clear and easy to understand.
You are highly encouraged to learn LaTex which is a very powerful tool for writing assignments,
essays and theses.

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Tags:
Sudoku Puzzle

row and column

possible numbers

four numbers

negation Laws

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