MATH PUZZLES 1-71

Daryl Cooper South Hall 6522 UCSB

The puzzles below vary in difficulty, generally later ones are harder. Almost all of them I have heard from other people. They are purely for amusement. I do not give out answers. I am always on the lookout for new puzzles. They should be unusual. Most routine puzzles are of no interest to me.

[1] An ant starts at one end of a rubber band and walks along it at a speed of 1 inch per second. The rubber band is 10 inches long, and is being stretched uniformly at a rate of 20 inches per second. Does the ant ever reach the other end of the rubber band?. Give reasons.

[2] Can you subdivide a cube into finitely many smaller cubes no two of which have the same size?

[3] What is the maximum number of knights that you can place on a chess board so that no knight threatens (ie can jump onto) any other knight? Can you show your answer is the best possible?

[4] Suppose I take a chess board, and remove two squares from opposite corners (ie the corners are on the same DIAGONAL) Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares?

[5] There are 12 balls, identical in appearance. One of the balls is either heavier or lighter than the others.If I give you a balance and allow you to make 3 weighings, how could you arrange to discover which ball is the odd one out, and whether it is heavier or lighter than the others? [using a balance, you can weigh balls on one side against balls on the other side]

[6] (Easier Prisoner Puzzle) A prisoner is held in jail.There are two doors in her prison cell, one leads to freedom, the other leads to death. There are two prison guards, one of which always answers questions truthfully, the other one always lies. Sadly the prisoner does not know which is which. She is allowed to ask one question of one of the guards, to which the answer must be YES or NO. What question should she ask in order to escape?.

[7] (Harder Prisoner Puzzle) Same set up as [6], except that there is only one guard, who sometimes lies, and other times tells the truth.

[8] 89 couples live on a Desert island. On this Island in paradise, the rule is that if a woman discovers that her husband has been unfaithful to her, then at noon on the day after she discovers, she shoots him dead. When a man is unfaithful, gossip spreads until every woman, except his wife, knows about it. Regretably, humans being what they are, every man has in fact been unfaithful. But all is well until one day a priest visits the enchanted Island for a year. He gets to know the people well. When he leaves, everyone is assembled to see him depart. He says how much he has enjoyed his stay, but that it has come to his attention that someone has been unfaithful. He does not say who, but hopes it will not happen again. What is the consequence of this remark if any?.

[9] In a TV show, a prize is hidden behind one of 3 closed doors. The contestant tries to guess where the prize is. After the contestant chooses a door, the host of the show (who knows where the prize is) opens one of the 2 remaining doors, to reveal that the prize is not behind that door. The host then gives the contestant the opportunity to change her guess. Should she?

[10] I have a bottle of vinegar and a bottle of oil. Suppose I transfer a spoonfull of oil to the vinegar bottle, and then partially mix the contents of the vinegar bottle, before transfering an equal volume from the vinegar bottle back to the oil bottle. Question: which is greater the amount of oil in the vinegar bottle, or the amount vinegar in the oil bottle?

[11] A monk walks up a mountain on monday. He sets out at dawn, the day is hot, and he has many short rests on the way up. He arrives at dusk. He prays all day Tuesday, and on Wednesday, gets up late, and jogs down the mountain getting to the bottom in the early afternoon. He follows the same route both days. Is there some place on his route which he comes to at EXACTLY the same time on both days?

[12] Can you find 1000 consecutive positive integers, so that none of them are prime?.

[13] Three marksmen are situated at random around a rapidly spinning sphere. They each shoot at the sphere and hit it. What is the probability that all 3 shots land in the same hemisphere?.

[14] Show that given any group of six people, you can always find either 3 people all of whom know each other or 3 people so that each of them does not know the other two.

[15] A race track has fuel dumps spaced around it. If all the fuel from all the fuel dumps is put into the tank of a racing car, it has exactly enough fuel to drive once around the track before running out of fuel. Suppose I give you a map showing you the location of the fuel dumps and how much fuel is in each dump. Your job is to try to find a starting position for the racing car (at some fuel dump), so that if you start driving around the track ,always taking the fuel from any dumps which you pass, you can drive once around the entire track without ever running out of fuel. Question: is this always possible?

[16] You are standing by a bicycle. If you push backwards horizontally on the pedal when it is at the bottom position does the bicycle initially move forwards or backwards?

[17] A cylinder which is completely full of water, except for a bubble of air held at the bottom, is sealed at both ends. There is no air gap between the top of the water and the top of the cylinder. The pressure the water exerts on the top of the cylinder is zero. Suppose the bubble is now allowed to rise to the top of the cylinder. Does the pressure at the bottom of the cylinder change? and if so, then by how much?. Assume that water is incompressible, and that the volume of the cylinder does not change.

[18] A gold brick in a row boat on a lake is thrown overboard. Does the water level in the lake rise or fall as a result of this?

[19] An iceberg floats in a lake, as it melts does the level of the lake rise or fall?

[20] Two people buy identical hot cups of coffee at the same time. The woman puts milk in her coffee immediately, the man waits 5 minutes and then puts milk in his coffee. The milk in both cases was the same temperature, and cold. If they both drink their coffee 6 minutes after buying it, who drinks the hotter coffee?.

[21] There is a cup half full of water on a scale. If you put your finger into the water without touching the sides of the cup, will the scale read a different weight?

[22] Suppose that you have a collection of 101 whole numbers which lie in the range between 1 and 200 inclusive. Show that there is at least one number in the collection which divides another one in the collection.

[23] Suppose you have a piece of wire 12 inches long, and that you cut it at two points chosen at random so that you now have 3 pieces of wire of varied lengths. What is the probability that you can make a triangle from the 3 pieces of wire without bending any of them?

[24] A 3x3x3 cube can be cut up into 27 1x1x1 cubes. If you are allowed to make planar cuts, what is the minimum of cuts you need to achieve this and why?. (you are allowed to move pieces around in between cuts).

[25] A mirror reverses left and right but not up and down. Why?

[26] There are two bottles each with a capacity of two liters. One bottle contains one liter of water, the other contains one liter of wine. There are two things you can do: (a) pour fluid from either bottle onto the floor, and (b) pour fluid from one bottle into the other. The only constraint is that you must always have at least one liter of fluid in the wine bottle. The object is to dilute the wine in the wine bottle with water as much as possible. What is the maximum dilution you can achieve?. Prove it!.

[27] How many prime numbers appear in the following sequence, where the digits are arranged in descending order? 9; 98; 987; 9876;.....; 987654321; 9876543219; 98765432198;......

[28] Can the product of four consecutive positive integers ever be the square of an integer?

[29] The Planet Vrog is divided into 8 countries, each of which occupies an octant. Thus each country borders 3 other countries. A traveler sets out on a trip to visit each country exactly once, returning to her home country at the end of the trip. How many different ways can this be done?

[30] What is the least number of knights which can be placed on a chess board so that every square is either occupied by a knight or else threatened by a knight.

[31] Estimate how much energy there is in all the wind over the entire Earth during one day.

[32] If you ride a bicycle along a straight road, is it more work to ride when there is a cross wind blowing at exactly right angles to the road than when there is no wind at all ?.

[33] A finite graph is a finite set of points called vertices in space together with a finite number of edges connecting pairs of vertices. Each edge meets exactly 2 vertices, one at each of its endpoints. If one edge meets another edge they meet only at one endpoint. Prove that if a graph has more than 1 vertex, then there are 2 vertices with the same degree (=the number of edges meeting the vertex).

[34] Given a positive real number x, one could take:

if x is too large then will grow without bound, ie What is the largest value that x can have without this happening?.

[35] 3 positive integers l,m,n are called a Pythagorean triple if : for example and . A Pythagorean triple is called Primitive if there is no integer greater than one which divides each of l,m,n, eg 6,8,10 is not primitive. How many primitive Pythagorean triples are there?

[36] There are 3 circles in the plane which all touch each other. No two circles have the same size. For each pair of circles draw the two lines which are tangent to these two circles. These two lines meet at a point somewhere. There are 3 such points, 1 pair for each way of choosing two circles. Prove that the three points lie in a straight line.

[37] Suppose you can manipulate a certain convex solid through a circular hole in a wall, is it always possible to first rotate the object into a suitable position, and then slide the object through the hole without having to rotate it any more?.

[38] A certain right-angled triangle has all its edges of integer length (eg 3,4,5). You can draw a circle in the triangle which just touches each of the 3 edges. Prove that the radius of this circle is an integer.

[39] I have an old Dodge station wagon. At the front of the engine compartment there is gap of about 6 inches called the crumple zone. During a collision, the crumple zone folds up, absorbing the energy of the impact. It is designed to absorb the energy from running into a stationary wall at 10 mph. Suppose I am driving on the freeway at 60 mph, and there is a brick wall driving in front of me at 50 mph in the same direction. I drive into it at a relative speed of 10 mph, so the crumple zone should absorb all the energy?. But wait a minute, the change in my kinetic energy is much greater in this case (because K.E. ). So maybe the crumple zone cant absorb all that energy?. Which is right?

[40] Two identical balls roll across the top of a table on parallel paths. One of the balls has to roll down into and up out of a dip in the table. The other ball rolls on the flat all the time. Which ball gets to the far side of the table first, and why?.

[41] A floor polishing machine has two circular brushes which rotate in opposite directions when the machine is turned on. Why is it easier to push the machine when it is turned on than when it is off?.(There is no wax involved.)

[42] Without calculation decide which is bigger, or

[43] Calculate

where [x] is the integer part of x ie the largest integer less than or equal to x.

[44] The game of reverse Tic-Tac-Toe, called Eot-Cat-Cit, has the same rules as Tic-Tac-Toe, except that the first player with 3 markers in a straight line looses. Can the player with first move avoid being beaten?.

[45] Given 5 points inside a unit square, show that at least two of the points are within a distance of each other.

[46] A right circular cone is rolled on a plane so that the apex V remains fixed. How many times will the cone revolve about it's axis if the cone is rolled thru a complete circle on the plane around V ?.

[47] A bicycle leaves tracks on the ground. The bicycle tires have no tread. Is it possible to decide which direction the bicycle was going?. (Sherlock Holmes claims he could do this in a certain story, which story?)

[48] What is the largest number of points on a sphere such that each pair of points can be connected by a path which does not pass thru any third point nor any other path?. What about on a torus (= surface of a donut)?

[49] If A is an complex matrix, when is there a matrix B such that : A=B.B, ie when does A have a square root?. If A has a square root, how many does it have?

[50] There are a countable infinity of spiders on the unit circle. The spiders can move at one inch per second, but must always stay on the circle. A fly can move at 2 inches per second, and starts out inside the circle. Can the fly escape from the circle without being eaten by a spider?. The spiders and fly are (of course) points.

[51] An infinite checker board has checkers below the x-axis on all black squares. Checkers are allowed to move by hopping over other checkers diagonally. The checkers which are hopped over are removed. How far up the y direction can a checker go?

[52] Two people play the following game by making moves alternately. If both people play perfectly who should win, the first or the second player?. There are N squares in a row, with 3 coins on the right most 3 squares. A coin can move to any un-occupied square on its left. The first person who cannot move loses.

[53] An electrical circuit is made up from resistors connected up in some complicated way. Suppose that you measure the resistance between 2 points in the circuit both before and after removing one of the resistors from the network. Is it possible that after removing the resistor, the resistance measured between the 2 points is decreased ?. Prove your answer carefully.

[54] Can you find uncountably many nested subsets of the natural numbers?. In other words are there uncountably many subsets of the natural numbers such that for every i,j either is a subset of or is a subset of .

[55] Consider a rectangular grid made up of squares. A path in the grid is a sequence of squares each connected either horizontally or vertically to the previous one. How many paths of minimum length are there from the bottom left corner to the top right corner?.

[56] A hydra is a rather fearsome finite tree G with a vertex called the root. Each vertex of G is connected back to the root by a unique path in G. The first vertex on this path after is called the parent of and the next vertex is called the grandparent of . If you form a new tree by cutting off a single edge e from G which is connected to the rest of G only at ie. the hydra grows a finite number of new copies of that branch of which starts at . If has no grandparent, then the hydra does not grow any new bits. Can you kill the hydra?.

[57] Given a bounded subset A of the plane, is it always possible to find 3 lines which meet at a point and so that each line exactly cuts A into 2 pieces of equal area?

[58] This is about a game called HEX. The plane can be tiled using regular hexagons, so that two hexagons either do not meet or else have an edge in common. Suppose that you cut out a roughly parallelogram- shaped region of the plane made up of some number of these hexagons. The top and bottom sides of this parallelogram belong to BLACK, the left and right sides belong to WHITE. Players move alternately, a move consists of putting a piece of your color on any unoccupied parallelogram. The winner is the first person to connect her two sides by a chain of pieces of her color. Show that a draw is impossible. Show also that if pieces are placed on the board until it is full, it is not possible for both players to have connected their sides

[59] How many disjoint copies of a figure eight ``8'' can be embedded in the plane, countably or uncountably many?.

[60] Is it possible to have two topological spaces X and Y which are not homeomorphic but so that each space is a finite sheeted covering space of the other?.

[61] There are two bottles each with a capacity of two liters. One bottle contains one liter of water, the other contains one liter of wine. There are two things you can do: (a) pour fluid from either bottle onto the floor, and (b) pour fluid from one bottle into the other. The only constraint is that you must always have at least one liter of fluid in the wine bottle. The object is to dilute the wine in the wine bottle with water as much as possible. What is the maximum dilution you can achieve?. Prove it!.

[62] Can you find a group G such that G is not isomorphic to the Automorphism group of any other group?.

[63] Some crackpot once wrote a book where he claimed that a comet came close to the Earth (but did not collide) and as a result the Earth's rotational axis was switched. In other words the Earth was rotated so that the positions of the north and south poles were changed. Is this in fact possible?

[64] There are two sealed envelopes containing money. You can choose one of them, open it and see how much money is inside. Then you have the choice of taking the money in the envelope you chose or else of taking the money in the other envelope without knowing how much is in it. Is there a strategy you can follow which will give you better than a 50% chance of choosing the larger amount of money?. Assume that there is a non-zero probability that there is between \$n and \$(n+1) in either envelope for every positive n.

[65] If you take a square and look at it from some point in space it looks like a quadrilateral. What are the possible shapes of this quadrilateral?

[66] A regular heptagon (= 7-sided polygon with all angles equal and all sides equal) is randomly placed in the plane. What is the probability that an observer can see 4 sides?

[67] A table has 3 legs of equal length. Is it always possible to place the table on a convex hill so that the surface is level?.

[68] By cutting a square along the 4 line segments joining vertices to midpoints of opposite sides, the square is decomposed into 9 pieces which can be re-assembled to form 5 congruent squares. Dissect a square into a "small number" of pieces by cutting it along straight lines, so that these pieces can be re- assembled to form 3 congruent squares. What is the least number of pieces this can be done with?.

[69] If you have a set A of 101 integers chosen from the numbers , show that every integer is the sum of 2 numbers from A modulo 200. (you can choose the same number from A twice)

[70] A lion chases a man inside the unit circle (they are allowed to go onto the circle as well). They both move at unit speed. Can the lion catch the man?.

[71] What is the maximum number of integer lattice points (= points with integer coordinates) in the plane that a strictly convex region can touch if it does not contain any lattice points in its interior.