Puzzle List -- CSCI 4900 -- Spr 03

Here is the list of puzzles for this class which gets you points for your overall score. There are no specific due dates, but after a while the point values go down because I'll start giving hints.

Keep up-to-date on this page in order to find out the current point value for a particular puzzle. (Hints are only given in class, on Mondays, so you needn't worry about checking multiple times per week.)

Often a puzzle has a long description and lots of details, so I will describe it in class rather than trying to draw pictures and post all the details here. But sometimes the puzzles are simple enough that they can be fully described on this page. The distinction is noted below.

On all puzzles, using a computer is allowed; for some it makes sense, for others it doesn't. Sometimes it is REQUIRED that you use a computer.

Note that I may post puzzles here before describing them in class, so be patient!!

Puzzle List

  1. There are three bins containing tennis balls, and the bins have open tops, but you cannot see into them because they are mounted too high on the wall. One bin contains just white tennis balls, another just yellow, and the third a mixture of white and yellow. There are three signs, one says "white", another says "yellow" and the third says "mixed" and these three signs are on the bins. However, you are told that no sign is on the correct bin. (Ie, they are ALL mismarked.) Your job is to draw one ball from a bin, look at it, then fix the signs so they are correct. (Points: 5)
  2. Solve the Indiana Jones problem (described in class). (Points: 10)
  3. Solve the "rope escape" problem (described in class). (Points: 15)
  4. Prove (yes, prove... as in math) that for any positive integer n there exists a positive integer m such that mn has only 0's and 1's in its decimal representation. (Points: 40)
  5. Write a program that solves the daily Quiddler puzzle at www.setgame.com. Many more details on how to do this were given in class. (Points: 150) (If you automate your solver to sign in at 1am (MST) and solve the puzzle without human intervention, you get a bonus of 40 more pts.)
  6. Solve the wooden block puzzle (distributed and described in class). You MUST use a computer to do this. No credit for "hand" solutions. (Points: 60)
  7. Solve the "two inverters" problem (described in class). You must use a computer for this; no "by hand" solutions allowed. (Points: 90)
  8. We will propose several "word problems" all of which will be based on the word list for this class. The first problem is the "longest anagrammed sequence" or "LAS" problem: find the longest list of words, w1, w2,... wk, such that w1 is any word and wi, 1 < i <= k is an anagram of wi-1 and any additional letter. For example, one valid sequence (though certainly not the longest) is: A, AM, MAN, MANE, NAMED, DEMAND, DEMANDS. This has a length of 7. (Points: 70)
  9. Here is a mathematical puzzle that you might use a computer to help you solve: consider a city street lined with n mailboxes, all initially closed. For the next n days, the following occurs: on day i, a boy a boy walks down the street and toggles the state of every i-th box (if it's open, he closes it; if it's closed, he opens it). Here's the puzzle: after n days, which mailboxes are open? You must PROVE your answer. (But it may help to simulate this on a computer to find what the answer is, then try and find a proof for it.) (Points: 40)
  10. Solve the non-linear ropes problems (described in class). Your goals are to time (1) Thirty mins with one rope (done in class), (2) 45 mins with two ropes (Points: 10), and (3) 20 mins with one rope (Points: 25)
  11. Solve the dotted-foreheads problem. (Points: 40)
  12. In the old days, Chicken McNuggets came in sizes 6, 9, and 20. What is the largest number of McNuggets you could not order? You may use a computer to help you find the answer, but you must justify that this is in fact the answer. (Points: 30)
  13. Prisoner's Dilemma: A warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another.
    "In the prison is a switch room, which contains two light switches labeled A and B, each of which can be in either the on or the off position. I am not telling you their present positions. The switches are not connected to anything.
    "After today, from time to time whenever I feel so inclined, I will arbitrarily select one prisoner and escort him to the switch room. This prisoner will choose one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell.
    "No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners arbitrarily. I may choose the same guy three times in a row, or I may jump around and come back.
    "But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.'
    "If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators."
    Devise a deterministic strategy for the prisoners that always eventually sets them free. (Points: 60)
  14. Solve the pool-table problem (described in class). (Points: 50)
  15. You have a black box, and embedded within it is some polynomial p(x) with degree n > 1 and positive integer coefficients. You know what n is, and you want to figure out what the coefficients are, but you cannot look inside the box. You are allowed to query this box with any integer i you like, and out comes the value p(i). Now, as you may or may not know, there is a fancy technique called "Lagrange Interpolation" which gives a way to find p(x) in n+1 queries. However in this case, I want you to find it in <= n queries. (Points: 30)
  16. Find the largest pancake number you can. (Described in class.) Points awarded will be based on the number you find.
  17. Solve the four-coins problem. Set-up: you are blindfolded, and set before you is a square lazy-susan with four coins, one on each corner. The coins start in any arbitrary pattern of heads-up or tails-up, and you do not know what this starting-pattern is. The table is always situated so that you are facing one of its four sides (ie, you never have a corner of the table in front of you). Your goal is to get all coins to be heads-up or all tails-up, subject to some rules.
    Rules: You are allowed to touch only two coins at a time, one with your right hand and one with your left. You are able to feel the surfaces of the coins when you touch them, and you can determine their orientation. Then you may flip zero, one, or both of them. After this, the table is spun by an adversary who is trying to make you fail. After the table is spun, you are given another turn with the same rules as above.
    If at any time you succeed (by getting all four heads-up or tails-up), a bell is rung and the game is over.
    No randomized solutions are allowed; your solution must work even against an adversary who knows your method, and must terminate in a reasonable number of steps. (Points: 30)
  18. Given any rectangle R which is tiled by smaller rectangles, show that if every sub-rectangle has at least one side whose length is an integer, then the same can be said about R. (Points: 80)
  19. Solve the following list of cryptograms using a computer program that you have written. Use techniques discussed in class and/or techniques of your own. (These were hand-typed so please forgive me if there are typos; my wife made a proofreading pass, but still there are no guarantees. Just keep this in mind if you have something that ALMOST makes sense but for one letter!) (Points: 80) And that assumes you solve ALL 20 of them.
  20. Below is the ciphertext produced with a program I wrote called vig.c. Note that this program disregards anything but uppercase characters. This means that punctuation, whitespace, and so forth, are all not represented in the ciphertext. (But rest assured that the plaintext was all uppercase, so it will be valid English when you decipher it.) Your assignment is to find the plaintext. Of course you will most likely want to write a program to help you with this task. But it's not a lot of coding: mine is 70 lines long or so. (Points: 50)
  21. This one is simple, but somewhat abstract: suppose you have two sets A and B which partition a subset of real numbers S. (This means that the intersection of A and B is empty and the union of A and B is S.) S is closed under multiplication. You also have the property that A and B are closed under products of three elements. (In other words, if you take any three elements from A, like a, b, c, then a * b * c is guaranteed to be a number in A. The same can be said about set B.) Prove that A or B must be closed under products of two elements. (Points: 25)
  22. Given a shape S constructed from any rectangle R which has a rectangular hole r inside of it, you can cut S into two equal halves with a single cut. Be sure and argue why your answer is correct. (Points: 15)
  23. Joe, a census-taker, arrives at Mary's door and asks how many children Mary has. She answers "I have three 3." Joe asks what are their ages, and Mary says, "well, I can tell you that the product of their ages is 72". Joe says, "that's not enough information to get their ages!" Well, Mary adds, the sum of their ages is the same as my address. Joe says, "That's still not enough information!" Finally, Mary has to go because her oldest child is sick, and she shuts the door. Joe, takes his lunch break and suddenly realizes he knows the ages of all three of Mary's children. What are they? (Points: 10)