What is the size of the largest collection of numbers from 1 to 100 with the property that no number in the collection is exactly twice as large as any other number in the collection?
What is the size of the smallest double-free collection of numbers from 1 to 100 which is complete, where by complete is meant that adding another number to the collection will cause some number in the collection to be twice as large as some other number in the collection?
Shortly after I thought of the previous problem I read about this one, whose statement and solution are closely related. It provides a nice contrast between the product of a hacker such as myself and true genius.
This problem seems monstrously difficult but it has a simple solution.
First an easy problem. Find 50 numbers from 1 to 100 such that none of them divides evenly into any of the others. There are many solutions to this. If there were not, the second part of the problem would be easy. There is, however, one solution that is fairly obvious.
Now show that 50 numbers is the best that can be done. That is, show that for any 51 numbers from 1 to 100 one of them must be divisible by one of the others.
Hint: Show that all numbers can be expressed as O*2n, where O is odd.
Use the insight gained from the previous demonstration to devise a scheme for finding the 50 smallest numbers up to 100 such that no number divides any of the others.
Given a 4 by 4 checkerboard, suppose that you can reverse the colors of any row or any column. How many patterns can you create by selecting different combinations of rows and columns. You might want start on a 2 by 2 checkerboard or a 2 by 1 checkerboard.
Note that squares belonging to the intersection of a selected row and a selected column may be thought of as being reversed twice, once by the column and one by the row, and so will end up with their original color.
a. Young Tom has become infatuated with Vera, a toll collector on the highway that Tom rides to work on. There are 3 tollbooths and Tom has learned that the toll collectors are randomly assigned to a tollbooth subject to the constraint that nobody works at the same tollbooth two days in a row. To simplify the problem assume that both Tom and Vera work 7 days a week. Assume also that Tom is unable to see who is working in any tollbooth other than the one where he is at. What strategy should Tom employ to maximize the chances of seeing the object of his affection? On average, how often will he see Vera?
b. This part is harder on both Tom and the reader. One day Tom notices Vera wearing an engagement ring. He is heartbroken, his only consolation being that Vera has never taken notice of him. He wishes to see Vera as little as possible. What strategy should he now employ? What are the chances on any particular day that the sight of Vera's countenance will take its toll upon Tom?
Consider the function f(x) = x2 + 1/x2 + x + 1/x for x > 0. For large values of x the x2 term dominates as the function goes to infinity. As x goes to zero the 1/x2 term dominates as the function again goes to infinity. It is reasonable to suppose that for some intermediate value of x the function attains a minimum value. To properly find this minimum value requires the use of calculus but given the information that the minimum is attained for a single value of x, a small insight into the nature of this function allows you to easily determine the value of x where the function has its minimal value. What is the value of x? What is the minimum value of the function?
A bus route has 14 stops. For each pair of stops there is exactly one passenger who gets on at the first stop and also gets off at the second. For example, there is one passenger who gets on at stop 2 and also gets off at stop 8. Find an equation for the number of passengers on the bus after it unloads and loads passengers at stop x. At what stop does the bus have the maximum number of passengers as it pulls away frmo the stop? How many people are on the bus at that point?
Using simple algebra, show that the 36th century (3500 to 3599) is the first century not to have a year that is an exact square. Hint: The following will prove useful: (x+1)2 - x2 = 2x+1
Find positive whole numbers that add to 12, for example 3, 4 and 5. Now multiply them together to get 3x4x5 = 60. Among all such sets of numbers adding to 12, which one gives the largest value when multiplied. You may get the answer by trial and error, but I think you will find the answer to be somewhat counter-intuitive. Can you generalize the problem and solution. Can you do some algebra to justify your answer?
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