A set of k positive integers is a postage stamp basis for n if every positive integer up to n can be expressed as the sum of no more than h values from the set. An extremal basis is one for which n is as large as possible. Design an extremal postage stamp basis (maximize n) with k=8 denominations such that every integer <= n … Continue reading the story "Postage stamp denominations"
Monthly Archives for April, 2013
Albert is looking at Betty. Betty is looking at Charlie. Albert is a computer programmer. Charlie is not. Is there a computer programmer looking at a non-computer programmer? Answer should be “Yes”, “No” or “Not Enough Information”. Hint: Don't rush, think for a while. via Math Factor
Player 1 thinks of a polynomial P with coefficients that are natural numbers. Player 2 has to guess this polynomial by asking only evaluations at natural numbers (so one can not ask for P(π)). How many questions does the second player need to ask to determine P? via Math Overflow Solution can be found here Hint: ... ... ... ... ... ... ... ... ... ... ... ... You can find the polynomial in 2 attempts.
There are 2 puzzles today. The easier puzzle first: Roll a penny around another fixed penny in the center with edges in close contact. After moving half circle around the center penny, you will find the penny in motion has rotated 360 degrees. Why? via CSE blog Now for the harder puzzle: Place four $1 coins as shown in the diagram below: Now roll the shaded coin … Continue reading the story "Rolling coins"
The game of Nim is played between 2 players A and B: There are 3 heaps numbered 1,2,3. Each heap contains 10 rings. Players take turns removing rings from heaps. In each turn, a player chooses a heap and removes as many rings as he wants from the heap. He cannot remove rings from more than one heap in a single turn. This continues until … Continue reading the story "Nim game"
Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle? Obviously the probability depends on how you choose the random chord. Suppose you require the following additional constraints for choosing how the chords must be chosen: Assume that chords are laid at random onto … Continue reading the story "Bertrand's paradox"
Does the equation, x^2 + y^3 = z^4 have solutions in prime numbers? Find at least one if yes, give a nonexistence proof otherwise. via Math puzzles
A group of students are sitting in a circle with the teacher in the center. They all have an even number of candies (not necessarily equal). When the teacher blows a whistle, each student passes half his candies to the student on his left. Then the students who have an odd number of candies obtain an extra candy from the teacher. Show that after a … Continue reading the story "Candy game"
Consider an n x n chessboard, where each square is arbitrarily chosen to be either black or white. Your goal is to make all squares in the chessboard white. At each step, you are allowed to "switch" a square, but each switch will toggle not only the particular square being switched, but also the 4 squares that are adjacent to it: Two vertically up and … Continue reading the story "Black and white squares"
A set J of integers is said to tile all integers if there exists a set C such that any integer x can be uniquely written as x=j+c where j is in J and c is in C. Intuitively speaking, J is said to tile integers if integers can be partitioned into disjoint tiles such that each tile is a translate of J. Show that if … Continue reading the story "Tiling integers"