Foundations of Data Science

Exercise 4.38 Consider a model of a random subset N(n, p) of integers {1, 2, . . . n} defined
by independently at random including each of {1, 2, . . . n} into the set with probability
p. Define what an “increasing property” of N(n, p) means. Prove that every increasing
property of N(n, p) has a threshold.
Exercise 4.39 N(n, p) is a model of a random subset of integers {1, 2, . . . n} defined by
independently at random including each of {1, 2, . . . n} into the set with probability p.
What is the threshold for N (n, p) to contain
1. a perfect square,
2. a perfect cube,
3. an even number,
4. three numbers such that x + y = z ?
Exercise 4.40 Explain why the property that N (n, p) contains the integer 1 has a threshold.
What is the threshold?
Exercise 4.41 The Sudoku game consists of a 9 × 9 array of squares. The array is
partitioned into nine 3 × 3 squares. Each small square should be filled with an integer
between 1 and 9 so that each row, each column, and each 3 × 3 square contains exactly
one copy of each integer. Initially the board has some of the small squares filled in in such
a way that there is exactly one way to complete the assignments of integers to squares.
Some simple rules can be developed to fill in the remaining squares such as if a row does
not contain a given integer and if every column except one in which the square in the row
is blank contains the integer, then place the integer in the remaining blank square in the
row. Explore phase transitions for the Sudoku game. Some possibilities are:
1. Start with a 9 × 9 array of squares with each square containing a number between
1 and 9 such that no row, column, or 3 × 3 square has two copies of any integer.
Develop a set of simple rules for filling in squares such as if a row does not contain
a given integer and if every column except one in which the square in the row is
blank contains the integer, then place the integer in the remaining blank entry in the
row. How many integers can you randomly erase and your rules will still completely
fill in the board?
2. Generalize the Sudoku game for arrays of size n 2 × n 2 . Develop a simple set of
rules for completing the game. Start with a legitimate completed array and erase k
entries at random. Experimentally determine the threshold for the integer k such
that if only k entries of the array are erased, your set of rules will find a solution?
Exercise 4.42 In a square n × n grid, each of the O(n 2 ) edges is randomly chosen to
be present with probability p and absent with probability 1 − p. Consider the increasing
property that there is a path from the bottom left corner to the top right corner which
always goes to the right or up. Show that p = 1/2 is a threshold for the property. Is it a
sharp threshold?
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