Here, in no particular order, are some of my favorite problems/puzzles. Many of them are from my MAA problem book, "Which Way Did the Bicycle Go?"
1. Given the tracks of the two wheels of a bicycle traveling through a mud patch, determine the direction of travel of the bike. Careful: Sherlock Holmes (The Adventure at the Priory School) got this one horribly wrong!
2. What is the rightmost nonzero digit of one million factorial?
3. (Victor Klee) For a set E in 3-space, let L(E) consist of all points on all lines determined by any two points of E. Thus if V consists of the four vertices of a regular tetrahedron, then L(V) consists of the six edges of the tetrahedron, extended infinitely in both directions. TRUE or FALSE: Every point of 3-space is in L(L(V))?
4. A tunnel underneath a large mountain range serves as a conduit for 1001 identical wires; thus, at each end of the conduit, one sees 1001 wire-ends. Your job is to label all the ends with labels #1, #2, . . . , #1001, so that each wire has the same label at its two ends.
You may join together arbitrary groups of wires at either end; they will then conduct electricity through the join. Then you cross the mountains by a very expensive and dangerous helicopter ride to the other end, where you can feed electricity through any wire and check which of the other ends are live, attach notes to the wires, and make (or unmake) connections as desired. Then you fly back to the near end, perform the same sort of operations, fly back, and so on as often as required.
How can you accomplish your task with the smallest number of helicopter flights?
5. A figure eight is a curve in the plane obtained from the basic “8” shape by any combination of translation, rotation, expansion, or shrinking; the lines forming the 8s are assumed to have no thickness. Is it possible to fit uncountably many disjoint figure 8s into the plane?
6. (A problem book by Friedland) Which planet is, on average, closest to Pluto? Assume circular orbits. Variations: Which planet is least likely to be closest to Pluto? Which planet is most likely to be closest to Pluto?
7. (A problem book by Friedland) Suppose you are playing poker with a small group of companions and a single deck of cards. If Lady Luck guarantees you a full house, and you can choose which full house you will get, which one should you choose? Hint: The answer is not "three aces and two kings".
8. Suppose we wish to know which windows in a 36-story building are safe to drop eggs from, and which will cause the eggs to break on landing. We make a few assumptions:
* An egg that survives a fall can be used again.
* A broken egg must be discarded.
* The effect of a fall is the same for all eggs.
* If an egg breaks when dropped, then it would break if dropped from a higher window.
* If an egg survives a fall, then it would survive a shorter fall.
* It is not ruled out that the first-floor windows break eggs, nor is it ruled out that the 36th-floor windows do not cause an egg to break.
If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carried out in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second-floor window. Continue upward until it breaks. In the worst case, this method might require 36 droppings. Suppose two eggs are available. What is the least number of egg-droppings that is guaranteed to work in all cases?
9. Alice and Bob ran a marathon (assumed to be exactly 26.2 miles long) with Alice running at a perfectly uniform eight-minute-per-mile pace, and Bob running in fits and starts, but taking exactly 8 minutes and 1 second to complete each mile interval (this refers to all intervals of the form (x, x+1), including, for example, the interval from 3.78 miles to 4.78 miles). Is it possible that Bob finished ahead of Alice?
10. (Lester R. Ford) Alice and Bob own roughly rectangular pieces of land on the planet Earth, which is assumed to be a perfect sphere of radius 3950 miles. Alice's land is bounded by four fences, two of which run in an exact north-south direction and two of which run in an exact east–west direction. Her north-south fences are exactly 10 miles long; her east–west fences are exactly 20 miles long. Bob's land is similarly bounded by four fences, but his north–south fences are 20 miles long and his east–west fences are 10 miles long. Whose plot of land has the greater area?
11. (Moshe Rosenfeld) A famous problem asks whether an 8x8 chessboard with two opposite corners deleted can be tiled with dominoes, where a domino is a rectangle congruent to two adjacent squares of the board. The answer is NO because each domino would have to cover one white and one black square, an impossibility since the number of white squares is different from the number of black ones. Now take an 8x8x8 cube with two opposite corners removed. Can it be tiled with 1x1x3 boxes (in any orientation)?
12. (Lee Sallows) Here is a self-enumerating crossword puzzle with a unique solution. Each of the six horizontal and six vertical entries is of the form, for example, "THIRTEEN NS". Here "THIRTEEN" can be any possible English number-word and "N" can be any letter in English. The idea is that if "THIRTEEN NS" is one of the entries, then the completed puzzle does indeed have exactly 13 instances of the letter "N". There are 12 entries, and so there will only be 12 different letters used in the completed puzzle. Every entry will have one blank cell, and an "S" occurs at the end when a plural necessitates it.
13. (Clifford Stoll) Suppose you are supervising three students, each armed with an inclinometer. You wish to place them in the plane so that they can determine the maximum height of a small rocket that you will launch after they are placed. When the rocket reaches its apex it will emit a flash, and at that time each student will measure the angle from the rocket to the horizontal through his or her position. So a student at point A will measure angle RAX where R is the rocket's position and X is the point on the ground underneath R. Is it possible to place the students so that, with the three angles that they provide to you, you can determine the height of the rocket (no matter where it is)? Note: The observers do not have compasses and get absolutely no information about the bearing (azimuth) to the rocket. All you get is three angles, and of course the positions of the students in the plane. The rocket veers in flight so that we have no information about its coordinates except that z > 0.
14. (Jeff Tupper) There is something about the graph of the following function that is totally shocking. What is it?
Graph the set of points (x, y) such that
but do so in the region 0 < x < 107 and k < y < k + 17 where k is the following 541-digit integer:
Created by Mathematica (February 27, 2004)