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?" Note that these are puzzles for which the solution is well-known, not unsolved problems. At the bottom of this list I will mention some of my favorite unsolved problems.
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
1/2 < Floor[ mod( Floor[y/17] 2^(-17 Floor[x] - mod(Floor[y],17)), 2)]
but do so in the region 0 < x < 107 and k < y < k + 17 where k is the following 541-digit integer:
960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719
15. (from Mathematical Mindbenders by Peter Winkler) A round cake has icing on the top but not the bottom. Cut out a piece in the usual shape (sector of a circle with vertex at the center), pull it out, turn it upside down, and replace it in the cake to restore roundness.
Do the same with the next piece; i.e., take a second piece with the same vertex angle, but rotated counterclockwise from the first one so that one boundary edge coincides with a boundary edge of the first piece. Remove it, flip it, and replace it. Keep doing this in a counterclockwise direction.
For some central angles what happens is clear. For a 90 degree angle it takes eight moves for all the icing to return to the top. If the angle is 180 degrees, it takes four moves to return to the initial state. Suppose the central angle is 181 degrees. When does all the icing first return to the top?
Also interesting: What happens if the central angle of each piece is 1 radian?
16. This problem is dedicated to the stalwart and skilled Macalester men’s soccer team. They were robbed!
Alice: Did you hear about the penalty kick shootout between the WorthWit and MacScot soccer teams?
Bob: I heard only that the MacScots shot first. What was the score? Who won?
Alice: The score was interesting. If I told you the score, you could tell me who won.
Who won?
Notes: A penalty kick shootout in soccer works as follows: There are five rounds of the form: Team A shoots; then team B shoots. However, if at any point the score is such that a team is guaranteed to win, the shootout ends. If they are tied after five rounds, then additional rounds are played until there is a winner. A score is reported as m-to-n where m > n (regardless of who shot first).
Postscript: writing down who has to know what for this to work,
Alice has to be smart (so that any computations she does are correct.)
You have to know that Alice is smart.
Alice has to know that Bob is smart, since she makes a statement about what Bob can do.
Therefore Bob has to be smart.
* In the fall of 2001 there was a very exciting NCAA Div. III soccer playoff game at Macalester. Whitworth C. scored in the 26th minute. Macalester equalized with 3 minutes left in the game. Nothing happened in overtime, so it went to a penalty kick shootout. Whitworth prevailed, after a controversial call by a linesman about the keeper’s movement.
17. A battleship of length 4 is moving along the real line in an unknown direction at a constant speed. It starts at an unknown position P, and is moving with speed V, but both of these are unknown real numbers. At every second you can shoot at some number on the real line; if you strike the ship the process ends successfully. Find a strategy that guarantees success in finite time.
18. Let S be the semicircle that is the upper half of the circle of radius 1 centered at (1,0). Consider also a circle C centered at the origin and let A and B be the points of intersection of C with the positive y-axis and with S, respectively. Extend the line AB rightward, letting X be its intersection with the x-axis. What happens to X as C becomes smaller and smaller, its radius approaching zero?
19. Alice and Bob are prisoners of warden Charlie. Alice will be brought into Charlie’s room on Sunday and shown 5 cards, numbered 1, 2, 3, 4, 5, face-up in a row in some arbitrary order. Alice can, if she wishes, interchange two cards. She then leaves the room and Charlie turns all cards face-down in their places. Bob is then brought to the room. Charlie calls out a random target card T. Bob is allowed to turn over ONE CARD ONLY and if, and only if, he finds the target T, the two prisoners are freed. The odds of success seem poor. What is the prisoners’ best strategy? Express the probability of success as n% where n is the nearest integer to the actual probability of your strategy.
Source: Stan Wagon, working with Larry Carter and Mark Rickert. Reference: The MENSA Correctional Institute by Carter and Wagon, American Math. Monthly, 2018.
The answer is very surprising. Indeed, this joins my list of problems that essentially everyone gets wrong. That list is, using the numbering here, #3, #15, #18, #19.
20. Take two identical glasses and fill them with tap water. Add a tablespoon of salt to one of them. Then place an ice cube in each. In which glass will the ice melt faster?
21. Due to Joe Buhler, Larry Carter, and Richard Stong. Consider a standard clock, where the hour, minute, and second hands (lengths h, m, s, resp.) all point straight up at noon and midnight. Are there positive integers h, m, and s and a time t such that the ends of the hands form the vertices of an equilateral triangle? This one is quite difficult. The answer is YES, but the smallest such clock has hands whose lengths are all near 10^600.
Favorite Unsolved Problem.
Consider some great circles on the sphere with no three meeting at a point. Form the graph whose vertices are the intersection points of the circles, and with two vertices connected by an edge if there is a simple arc between the two points on the sphere. This is called the arrangement graph of the great circles. Conjecture: Such a graph is always 3-colorable.
Deleted. Matt Cook things it follows by the same method as for handles.
2. Suppose we have a polygon in the plane. Suppose we have two points A and B defining a line segment initially positioned well away from the polygon Suppose the line segment can be moved around the polygon. So the segment AB defines a ring that can pass over the polygon. Now suppose B is moved so that the opening is wider. Is it still the case that the polygon and its handles can fit through the two points? Reference: Problem III and Theorem 4 of https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Goodman-Pach-Yap494-510.pdf