On this page you will find a collection of problems in math and logic. You can solve these problems and compete with other visitors to The Truth Tree. Each problem has a point value. Your score will be the sum of all the point values of the problems you have solved correctly less a penalty for wrong answers. You may try repeatedly to get the right answer, but each time you miss a problem a point is deducted from your score. The first thing to do to participate on this Board is to sign in. You don't have to use your real name, but if you want your score to build up you will have to use the same name and password every time you log in. You must also enter a password so that your answers will be credited only to your name. Without the password, some practical joker could enter a lot of wrong answers under your name, thus decreasing your score.
There are two scoreboards. The first one lists the contestants in order of their scores with the top scoring contestants listed first. This one can be found at The Main Scoreboard. The second lists the contestants alphabetically and shows which problems each contestant has solved or missed. This scoreboard is The Itemized Scoreboard. Be sure to check this second one to ensure that the name you have chosen has not already been used by another contestant. Solve as many problems as you like each time you log in, but please wait until you are finished solving problems before clicking a Submit button. For convenience, there is a Submit button at the end of each problem, but clicking on any one of these will submit all answers you have given.
The scoreboards are updated daily when possible. Don't expect your answers to appear immediately after you have posted them!
Now look in the index below and click on a problem to solve. Or, alternatively, you can simply scroll down through the problems. The answers to some problems will be in words. Other problems have numerical answers. Give the numerical answers to four significant figures when appropriate. Numerical answers that are whole numbers can be entered as whole numbers. It is not necessary to include units. If the answer is "3 feet" the number "3" is sufficient. If you have a question about a problem leave your email address in one of the answer boxes with a brief description of your question and Remi will get back to you.
You might also want to check this site for more interesting puzzles. And go to this site for interesting IQ & Personality Tests. Also an index to numerous similar sites can be found at Gamestalks.
Index
An army one mile long starts to march at a uniform rate. It marches exactly one mile and then stops. At the same instant that the army starts, a messenger, who is at the rear rank, also starts to march at a uniform rate. He marches alongside just fast enough to reach the front rank, turn around instantaneously, and return to the rear rank again at the exact moment the army stops. His rate of marching and the army's rate of marching are uniform throughout. How far did the messenger walk?
Answer: Beware! This problem is harder than it looks! Value: 20 points.
If a steel ball one inch in diameter weighs one pound, how much will a steel ball two inches in diameter weigh?
Answer: Value: 5 points.
Back in the glorious days of communism in the Soviet Union, the glorious people's telecommunications committee came up with a glorious invention. It was a fiber optics cable which circled the earth and which, the committee solemnly promised, would revolutionize worldwide telecommunication. Because of technical considerations beyond the scope of this (or any other) problem, the cable could not be bent. Not even a little bit. Just how they managed to get it through mountains and stuff is also beyond the scope of this problem. Such questions are forbidden. Anyhow, it was a perfectly circular cable. Unfortunately it went right through a certain chicken farm on the outskirts of Minsk. The farmer complained that his chickens refused to cross over the cable. He tried tunnels and bridges and the chickens refused both. The easy solution was to lengthen the cable just enough so it would clear the ground by one foot. Of course, this meant that it would also have to be one foot off the ground everywhere else on earth. How many feet must be added to the length of the cable to bring about this glorious result?
Nancy is a very attractive clerk in a candy store. She is 19 years old, is a Freshman at an Ivy League University, wears fashionable clothes, and is a highly competent mathematician. She has done a least squares analysis on her weight during the last two years and has found that the best fitting polynomial is a cubic equation. What does she weigh?
All, who neither dance on tight ropes nor eat penny-buns, are old. Pigs that are liable to giddiness are treated with respect. A wise balloonist takes an umbrella with him. No one ought to lunch in public who looks ridiculous and eats penny- buns. Young creatures who go up in balloons are liable to giddiness. Fat creatures who look ridiculous may lunch in public if they do not dance on tight-ropes. No wise creatures dance on tight-ropes if they are liable to giddiness. A pig looks ridiculous carrying an umbrella. All, who do not dance on tight-ropes and who are treated with respect, are fat.
Therefore, no wise young pigs... Value: 20 points.
As I was going to Saint Ives, I met a man with seven wives. Each wife had seven cats, each cat had seven kittens. How many were going to Saint Ives?
Mr. Smith and Mr. Jones were talking one day while sitting on Mr. Smith's front porch. A third man joined them. Mr. Smith said, "I could introduce you to this man, but it might be more fun for you to try and figure out who he is. Brothers and sisters I have none, but this man's father is my father's son. Who is this man?" Mr. Jones answered correctly, saying to Mr. Smith, "Well, he must be ___________________."
Note: The correct answer to this problem must finish Mr. Jones's sentence correctly.
Fill in the last two numbers.
1, 3, 9, 4, 12, 144, 139, ..., ...
Be sure to use a space between the two numbers you choose. That is to say, if you think the answer is 234 and 1765, type the following:
Back in the 17th Century, Blaise Pascal ran a pizza parlor in Paris where three main roads cross to form what was called in those days Pascal's Triangle. And of course all the pizzas were triangular in shape. Not many people are aware of these facts, but one of the advantages of calling The Truth Tree is that you will find out little known but interesting facts about life and the world around you. <ahem> Anyway, being of a probabilistic turn of mind, old Blaise would sell you a pizza with any combination of 10 different toppings all the way from no toppings to all 10. He wouldn't allow double (or more than double) helpings of any of the toppings. How many different pizzas could you order?
Once there was an old lady who brought a basket of eggs to market. A horseman came by and the horse stepped on her basket and broke all the eggs. The horseman offered to pay and asked her how many eggs had been in the basket. The old lady said, "Well, I'm just a poor old woman and can't count above 7. But when I counted them 2 at a time I always had one left over. When I counted them 3 at a time I had one left over. And the same thing happened when I counted them 4, 5, and 6 at a time. But when I counted them 7 at a time it came out even." What is the smallest number of eggs she could have had?
Answer: Value: 10 points.
Between points A and B there are two railroad tracks. One of them is straight and is 4 miles long. The other one is the arc of a circle and is 5 miles long. What is the radius of curvature of the curved track?
Answer: Value: 20 points.
Between two buildings there are two ladders leaning criss-cross. One ladder is 8 feet long and the other is 5 feet long. The base of each ladder is on the ground against one of the buildings with the other end resting on the opposite building. Assume that the ground is perfectly flat and horizontal and that the buildings are perfectly vertical. The vertical distance between the point where the two ladders cross each other and the ground is 3 feet. How far apart are the buildings?
You have four cards on the table. You know that each card has a letter on one side and a number on the other. There is a rule that says, "If a card has an M on one side it must have a 3 on the other." Which card or cards of those shown below would you have to turn over to make sure that the rule was not violated for any of these four cards?
To answer, type the letter of each card you think should be turned over. (For example, if you think that cards B and C should be turned over, type "BC" in the answer box.)
You are the bouncer in a bar. It is your job to see that no one under the age of 21 is drinking beer. All the customers have cards hanging around their necks. On each card their age is printed on one side and what they are drinking is printed on the other. Of the following four cards, which card or cards would you have to turn over to make sure that none of these four customers was breaking the law?
We have a regular tetrahedron whose faces, instead of being plane surfaces, are spherical in shape with the center of each sphere at the corners of the tetrahedron. This solid has been called a "spherical tetrahedron" and has the interesting property that it (almost) acts like a sphere. That is, when it is on a plane surface, the highest point above the plane is at (almost) constant height. One can place three of these on a table and lay a bread board (or an ouija board for that matter) on top. The board can now be moved about in the horizontal plane and the tetrahedra will act like balls. The present problem has to do with the intersection of any two of the spheres. A little thought will show that their circle of intersection is not a "great circle" on either of the spheres. It is some smaller circle resembling a parallel of latitude. The question is: what is the latitude of this circle?
What is the volume of a regular tetrahedron with edges one unit long?
Everyone at a party shook hands with everyone else exactly once. In all, 66 handshakes took place. How many people were at the party?
(This problem was suggested by Enchantress.)
There was a bear photographer who left his tent and walked a mile south. He then turned and walked a mile east and turned again and walked a mile north and found himself back at his tent. He saw a bear. What color was the bear?
(This problem suggested by Kevin Huang.)
Suppose that you have a table with 8 rows and 8 columns and that the numbers from 1 to 64 are placed in the table in such a way that all the rows and columns add up to the same number. What is this number?
(This problem was suggested by Joy Vong.)
You have a can that is 3 inches high and 4 inches in circumference. Point A is a point on the upper rim of the can. Point B is a point on the lower rim directly below point A. That is, the line AB is perpendicular to the table on which the can is resting. A woggle bug starts at point A and walks by the shortest path to point B that goes around the can one time. How far did the woggle bug walk? Another way of looking at this problem is to ask how long a string would have to be so that it could go around the can once with one end at point A and the other at point B.
(It isn't necessary to know what a woggle bug is.)
One day George went to his favorite liquor store and bought a bottle of his favorite whiskey. He had bought liquor at this store for many years and was well known to the sales clerk. As he was paying for his purchase he happened to remark, "You know, today I celebrated my 18th birthday." How old was George?
(This problem was suggested by Kevin Huang.)
There was once a fellow named O. C. Disorder who was a bit long of leg and short of foot. The soles of his shoes were, in fact, exactly 9 inches long and his stride was exactly 35 inches. He had a habit of counting his steps when they were all on the same slab of the sidewalk and saying "Boing!" every time he stepped on a crack. If he stepped over a crack his counting started again at one, and of course his counting started at one after each "Boing!" In his neighborhood there was a sidewalk with perfectly regular slabs all the same size. He noticed that when he walked along this sidewalk he always got the following repeating pattern where "*" stands for "Boing!" 121231231212312312*121231231212312312*121231231212312312*121231....et c. How far apart, in inches, were the cracks in the sidewalk?
(This problem was invented by Remi who always says BOING! when he steps on a crack.)
John is twice as old as Susan was when John was as old as Susan is now. The sum of John's and Susan's ages is 28. How old is John?
Once there was a man who had some eggs in a box. He met a friend and gave him half the eggs plus a half of an egg. Then he met a second friend and gave him half the remaining eggs plus a half of an egg. Finally, he met a third friend. When he gave the third friend half the remaining eggs plus a half of an egg he had no eggs left. How many eggs did he have to start with?
In the figure above AC=BD. Angle ACD=90º. Angle BDC is less than 90º. QP and RP are perpendicular bisectors of AB and CD respectively. They intersect at P. AP=BP and CP=DP because all points on the perpendicular bisector of a line are equidistant from the extremities of the line. AC=BD because this is given. Triangle APC is therefore congruent to triangle BPD because of the well known "side-side-side" theorem. Now, Angle PCD=Angle PDC because the base angles of an isosceles triangle are equal. Angle ACP=Angle BDP because they are corresponding parts of congruent triangles. It is now perfectly clear that Angle ACP + Angle PCD = Angle BDP + Angle PDC because when equals are added to equals the results are equal. We have thus demonstrated that Angle ACD=Angle BDC which is contrary to fact. This problem led to a famous theological argument. A mathematician had a rickety table (with top AB sitting on the floor CD with legs AC and BD) and had constructed the drawing above in an ineffectual effort to straighten the table. He explained the dilemma to his friend, who was a theologian. The theologian said, "Well, I'm not surprised that your reasoning led to a contradiction. You will never arrive at the truth through mathematics but only through reliance on the Word of God!" If you can figure out what is wrong with this proof, explain the error in the box below. A correct refutation is worth 20 points and may also help straighten your table! However, I doubt it will save your soul!
There was once a mathematician named Afine Plane who was down on his luck. The only job he could get was one working as a roustabout in a circus. One day the owner of the circus (who happened to be named Nicolai Ivanovich Lobachevsky) remarked that the top of the tent was either infinitely far from the ground or else parallel lines intersect in a finite distance. He knew that Afine was a mathematician, and he wanted to show off his own mathematical sophistication. He took Afine into the circus tent and pointed out to him that there were two main poles which held up the tent. They were indeed very tall and they extended upwards and vanished out of sight. They were made of a special graphite fiber material which could not bend and were therefore perfectly straight. They were exactly 100 feet apart at ground level. Now Afine had helped to put up the tent and he knew that the tops of those two poles were fastened together at the top of the tent. Nicolai invited him to check whether the poles were in fact vertical, and careful measurement revealed that they were indeed perpendicular to the ground. "But that's impossible!" cried Afine. "From a point to a plane one and only one line can be drawn perpendicular to the plane!" "Not necessarily," replied Nicolai. "Come. Let me show you." So he led Afine to his tent where, much to Afine's surprise, he had a blackboard. Nicolai then drew the following diagram on the board.
He said, "AP and BP are the diameters of two spheres. The points A and B lie on a plane. Since the two diameters meet at point P the two spheres dip into the plane creating two intersecting circles, as shown. The points of intersection of these circles are Q and R. Now note that angles PQA, PQB, PRA, and PRB are all right angles because they are inscribed in semicircles. It is therefore clear that PQ and PR are both perpendicular to the plane since if a line is perpendicular to two lines in a plane it is perpendicular to the plane." After delivering himself of this discourse, Nicolai went to bed very happy because he was sure he had put one over on a mathematician. His victory so enhanced his self esteem that his wife wondered what had so improved his sexual performance. However, he wasn't so happy the following morning when he saw Afine who said, "Your proof is flawed, and here's what's wrong with it." What did Afine tell Nicolai? (20 points.)
Once there was a king who had a beautiful son named Alexias. A prince named Lysis from a neighboring kingdom fell in love with Alexias and asked the king for Alexias's hand in domestic partnership (with full medical benefits.) The king (no homophobe he) agreed on the condition that Lysis succeed in a guessing game. On one side of the grand ballroom of the palace were three doors. One of these led directly to Alexias's quarters, one was a mop closet, and one led to a stairway down to a dank dark dungeon in which Alexias's curiously misshapen younger brother did cruel things to captive cockroaches. Lysis was instructed to choose one of the doors without opening it. The king then opened one of the other doors, thus revealing either the mop closet or the entrance to the dungeon. He then said to Lysis, "If you can guess which of the two closed doors leads to Alexias's quarters you may take him away to your castle and live happily ever after. You can either stay with your first choice or switch to the other door, whichever you think gives you a better chance of guessing correctly." Assuming that Lysis uses the best strategy, what is the probability that he will choose the right door? What is the probability of living happily ever after? (Don't answer that last question!)
One year Natasha gave Boris a beautiful new suitcase so that when he attended the Soviet Mathematics Association in Moscow he would look just as sharp as all the other mathematicians. (You may or may not have an idea just how sharp a typical Russian mathematician looks.) Unfortunately, Natasha was a very jealous wife and sometimes convinced herself on very little evidence that her husband did more than mathematics at these annual meetings. This year, when he returned from Moscow, she noticed that his beautiful new suitcase had several perfectly straight scratches on one side of it. She immediately became very suspicious and started giving Boris the third degree. He explained to her that while he was waiting for his suitcase to come around on the conveyor belt in the airport a cat, who had been frightened by a Norwegian elk hound, came sailing through the air and landed on his suitcase with claws outstretched. It seems that the suitcase was traveling due north at 3 kilometers per hour and the cat was traveling due east at 5 kilometers per hour when it landed on the suitcase. When it landed, it naturally slid, thus making the scratches. Natasha was furious! "I've caught you in a lie!" she screamed. "Those scratches would be curved if they were made by that cat! I know you're just trying to cover up the truth!" Which of the following correctly describes the scratches that would appear on the suitcase if indeed they were made by the cat under the given circumstances?
A. Curved because the belt and the cat were going at different speeds. B. Curved because the cat would decelerate while sliding across the suitcase. C. Curved due to the Coriolus force. D. Straight because the deceleration vector would be colinear with the velocity vector. E. Straight because the motion of the suitcase was at right angles to the motion of the cat. F. Straight because the Coriolus force would not operate on a body moving due east.
(Enter the letter of the correct answer:)
One day in math class Nerdly G. Obstreperous was being his usual obstreperous self. His teacher, Kindly J. Wisdom, gave him the following problem to quiet him down. "Nerdly," he said, "You did an excellent job on problem number 18 about the photographer who saw a bear. Remember that he started from his tent and walked a mile south, then a mile east, and finally a mile north and found himself back at his tent. This seems impossible at first glance, but as you correctly pointed out it is possible under some circumstances. But can you give me a complete description of all the ways it can happen?" Nerdly replied, "You mean all the places on the planet where it could happen? Well, you know there's only one place it can happen, but I'd better not mention the location because it might give an innocent visitor to The Truth Tree an important clue as to the solution of problem 18." Mr. Wisdom smiled benignly and gave his characteristic (and infuriating) chuckle which went something like, "Hng-hng-hng..." Then he said, "I'm afraid you've got that one wrong, Nerdly. Actually there's an infinite number of places where it could happen. If you can tell me the answer before the end of the period I'll give you 20 points on your last test." Of course Nerdly got very quiet and remained that way until just before the bell, thus proving yet again how kindly and wise this math teacher really was even if he did have an annoying chuckle. Suddenly Nerdly jumped up and said, "Oh!! I know where all those other places are!" Then he revealed what he had discovered. If you can give the specification for all those other places in the text box below, you can add 20 points to your score!
Once there was a college student whose father was very generously paying not only his tuition but also his living expenses. : Whenever this student needed money he usually wrote his father a nice letter in which he explained how he had attempted to economize but that he was out of funds. He was typically rather apologetic about asking for more money because he feared that his father's patience would wear thin eventually, and then life wouldn't be so easy anymore. But one day he was in a hurry and merely sent his father an e-mail that said, "SEND MORE MONEY." When his father received this rather peremptory demand he decided that his son needed to be taught a lesson. He sent his son a letter which contained the following:
He explained that this was an addition problem in which each letter represented a single digit. He promised to send the amount of the total if his son could first figure out what that total was. What was this total amount of money?
A theater has 100 seats. Men must pay a dime to enter. Women are charged a nickel. Children are admitted two for a penny. One evening, the house was full and the total receipts were one dollar. How many men, women, and children were there? To submit your answer type in the number of men, a space, the number of women, another space, and the number of children. (This problem was suggested by Ronnie Sims.)
In a university town there was a circular court with ten houses in it. It was called Math Circle because so many math professors lived there. The houses were numbered from zero to nine, inclusive. One of the mathematicians in residence noticed that there was a rule which determined the number of mathematicians living in each house. The rule states that the number of mathematicians living in a given house is equal to the number of houses containing a number of mathematicians equal to the given house's street number. For example, 5 mathematicians in house 2 means there are 5 houses with 2 mathematicians in each. So, now that the problem has been clearly stated (!?) can you tell how many mathematicians lived in each house? Express your answer as a ten-digit number in which the first digit tells how many mathematicians live in house 0, the second how many in house 1, etc. (Warning! Prolonged contemplation of this problem may cause brain damage!)
(This problem was suggested by Ronnie Sims.)
Remi was playing black against Ron, who was playing white. Remi found himself in a perilous position to say the least, but hope springs eternal in the chessplayer's heart, especially when he is even or better in strength. He was still hoping to win. It was Ron's turn. To Remi's dismay Ron excitedly announced "Mate in Two!" and confidently turned off the clock. See if you can find the moves that lead to this quick win. (Credits: William Barclay, 1st Pr. USPB, 1966)
(Please enter your solution in algebraic notation. Go here for instructions if needed.)
Ron grew overconfidant, as often happens to victorious chessplayers. In the next game he decided to give Remi a handicap by firing up his water bong! Waving away the second-hand smoke, Remi applied the full force of his miraculous mental powers and reached this position with the White pieces. If you can work out a win for White in 9 moves, you will earn double the points offered for the previous chess problem. (Credits: Volker Zipf, 3rd Pr. Macleod MT, 1994) We capped the number of chess problems that increase your score to two, but if you want to try one more just for fun (no points), click here and a new window will open up.
(This problem was suggested by Eamon Warnock.)
Value: 10 points.
Mr. and Mrs. Smith had a dinner party and invited four other couples: the Blacks, the Clarks, the Joneses, and the Whites. When all the guests arrived there were thus ten people in the house. Some of these people had met before and some hadn't. All the people who had never met shook hands. Then Mr. Smith asked every guest and his own wife how many hands each of them had shaken. To his surprise every person gave a different answer. How many hands did Mrs. Smith shake?
(This problem was created by A. R. Legard and was suggested for posting on the Tree by Bill Gretton.)
A remote tribe makes an annual levy on each male member equal to one pound weight of grain for every year of his age. Traditionally each contribution of grain is weighed on a balance scale using at most three of the tribe's seven ceremonial stones. Each stone weighs an integral number (a whole number) of pounds and the stones used do not have to be placed on the same side of the balance scale. If ever a tribesman lives to such an age that his contribution cannot be weighed according to tradition, the levy will be abolished and no further taxes can ever be collected. Consequently the stones have been selected so that this age is the maximum possible. The weights of these stones were carefully selected by an ancient chieftain whose name has been forgotten. He was, however, obviously a formidable mathematician! What are the weights of the seven stones? (Give your answer as seven numbers separated by a space, e.g., "3 12 26 41 68 75 93".)
Value: 20 points.
(This problem was suggested by Gerald Brown.)
Given four points in 3-space, not all in the same plane, how many different planes are there that are equidistant from all four points? (If this doesn't stretch your brain, report immediately to The Institute for Advanced Study at Princeton.)
(This problem was suggested by Lloyd Milligan.)
Back in the good old days in a certain country, the mail service was considerably less than ideal. In fact, it was downright terrible! Postal employees were so dishonest and the bureaucracy was so inefficient that anything put in the mail would be stolen if it was valuable and trashed as junk mail if it wasn't valuable. To solve this problem the Postal Service sold special boxes which could be reused any number of times. The boxes were very sturdy and were equipped with hinged lids secured by large, stout hasps large enough to accomodate more than one padlock for those who wanted to be extra secure. They could not be opened when padlocked without the key, and it was illegal to break into them by any means. The padlocks were available for purchase from the Postal Service, and only padlocks sold by the Postal Service could be legally used. There existed exactly one key for each lock and one lock for each key, and it was illegal to copy keys or locks. It was also illegal for anyone to have in his or her possession a key belonging to someone else. Combination locks were illegal. In short, nothing could be legally mailed that was not in a regulation box, and only regulation padlocks could legally be used to lock those boxes. An unlocked box would be trashed, and its contents would be confiscated if they were valuable and thrown away if they were not valuable. Now it so happened that Boris, who lived in St. Petersburg, had to send Natasha, who lived in Vladivostok, a diamond necklace by mail. How could he do this without anything illegal being done by anyone?
The Great Detective found the following image at the scene of a murder. Using his incredible powers of deduction he correctly concluded that the name of the murderer was somehow encoded on this image. Can you figure out who the murderer was? Fill in his name in the box below to solve this puzzle. (This single image random dot stereogram was created by a program written by Remi.)
Looking at the image below in a special way you may see a solid figure. How many different faces (or surfaces) does it have? Another way of asking this question is, how many bugs can you place on this object so that none of them can meet any other bug without crossing or reaching across an edge?
A painting contractor was asked to estimate the cost of painting the floor of a merry-go-round. The polyurethane paint he wanted to use was very expensive, and he needed a very accurate measurement of the area to be painted. He sent his assistant with a steel tape measure to get the required information. When the assistant arrived, however, he was frustrated because his tape wasn't long enough to go all the way around the circumference of the merry-go-round, and he couldn't measure the diameter or the radius because the circular hole in the center of the merry-go-round floor was filled up with the calliope and all the machinery for making the merry-go-round turn. So he fastened one end of his measuring tape to an arbitrarily chosen point on the outer edge of the merry-go-round floor and walked across to the other side. Then, keeping the tape tight and straight, he moved along the opposite edge of the floor until the tape just grazed (was tangent to) the central circular hole and found that the distance from where he had initially fastened the tape to the opposite edge while the tape was tangent to the inner circle was 70 feet. When the assistant returned the contractor was at first furious with him. He said, "How in the world can I calculate the area of that floor with only that one measurement?" The assistant, who was pretty smart, explained how it could be done. What is the area of the floor of the merry-go-round?
A machinist drilled a round hole straight through the center of a solid spherical steel ball. It went all the way through the ball, and the hole was exactly 6 inches long. How many cubic inches of steel were left in the finished product? It may seem that not enough information is given here, but a little thought (or maybe more than a little) will reveal that the exact volume can be found.
This problem has been stated in the form of a poem, as follows:
Old Boniface he took his cheer, Then he bored a hole through a solid sphere, Clear through the center, straight and strong, And the hole was just six inches long. Now tell me, when the end was gained, What volume in the sphere remained? Sounds like I haven't told enough, But I have, and the answer isn't tough!