The Contest Center - Tough Puzzles

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Tough Puzzles
The Contest Center
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We will post the names of anyone who solves any of these puzzles.

OMG Equation
Solve for x: √(2x-2) + √(25-4x) + √(100-16x) = √(67-12x) + √(43-8x) + √(24x-29)
Solve algebraically, not numerically, to get an exact solution where x is a positive real value. Please explain your method. (It is not necessary to show all of the steps.)

Solved by: Arthur Vause

Unknown Triangle
       Three university students, A, B and C, applied for a job in the computer laboratory. All were straight-A students in both math and computer science. Since there was more than a month before the school year started, I devised a test to choose the best candidate.
       On Day 1 I met with the 3 students and carefully explained the procedure. I had chosen a certain Pythagorean right triangle with integer sides. The hypotenuse was between 300 and 2300, inclusive. I would secretly tell A the length of the shorter leg, B the length of the longer leg, and C the length of the hypotenuse. Then I would ask them, in turn, if they could tell me the lengths of all 3 sides, giving them a full day each time to consider their answers. They would all hear the other students' answers.
       On Day 2 I assembled the 3 students and asked A what are the lengths of the 3 sides. A did not know.
       On Day 3 I assembled them and asked B for the lengths of the 3 sides. B did not know.
       On Day 4 I asked C for the lengths of the 3 sides. C did not know.
       On Day 5 I asked A again for the lengths of the 3 sides. A did not know.
       I continued asking them in turn for lengths of the 3 sides. Finally, on Day 17, after I had asked A for the sixth time, and A did not know, both B and C shouted out "I know!"
       What is the length of the shortest side?

Solved by: Mark Mammel, P.M.A. Hakeem, Ender Aktulga

Beehive
Write the numbers from 1 to 19 into the cells of the beehive so that every row and every diagonal line adds to the same total.

Solved by: Gilles Ravat, Jean-Charles Meyrignac, Sudipta Das, Saurabh Agarwal, Andreas Abraham, Gaurav Agrawal, Mark Mammel, Bert Sheldon, Mark Thornquist, Terje Kristensen, Don Dechman, Jesse Wiedman, Mark Rickert, Anupam Mittal, Rupa Kamat, P.M.A. Hakeem, ace03, Marvin Cervania, Paul Cleary, Naim Uygun, Ionut-Zaharia Chirila, Ender Aktulga, Kaustuv Sengupta

181 (Submitted by Paul Cleary)
Find a non-trivial integer solution to the equation x² - 181y² = 1.

Solved by: Mark Rickert, Naim Uygun, Ionut-Zaharia Chirila

Partition (Based on a problem from Paul Cleary)
How can the set of consecutive integers from 5 through 68 be partitioned into two equal-sized subsets whose sums, sums of squares, sums of cubes, sums of fourth powers and sums of fifth powers are all equal?

Solved by: P.M.A. Hakeem, Mark Rickert (3 solutions), Ionut-Zaharia Chirila, Ahsen Canat, Arthur Vause (3 solutions), Ender Aktulga

Bases
I have a program on my computer that can convert a number from any base to any other base. I started with some number, converted it to another base, and the result was one digit shorter. Then I tried a different base, and the result was again one digit shorter. Altogether I did this 10 times, and the number grew one digit shorter each time, so the final number was 10 digits shorter than the starting number. (The final number had more than 1 digit.) What is the smallest number I could have started with?

Solved by: Sean Kickham, P.M.A. Hakeem, Nick McGrath, Mark Rickert

Piling Stones
Tom, Dick and Harry piled stones into a pyramid to my precise specifications. Fred, an efficiency expert carefully watched them. When the pyramid was complete, Fred reported that if Tom had worked twice as fast the job would have taken exactly 2 hours less, if Dick had worked three times as fast the job would have taken exactly 3 hours less, and if Harry had worked four times as fast the job would have taken exactly 4 hours less. How long had the job taken?

Solved by: P.M.A. Hakeem, Paul Cleary, Nick McGrath, Ionut-Zaharia Chirila, Mark Rickert, Kiran Limaye, Naim Uygun, Ender Aktulga, Kaustuv Sengupta

200 Candies
I have 200 candies in an hourglass dispenser. There are 40 each of red, white, blue, yellow and green. Each time I turn the hourglass over the candies get thoroughly mixed and 20 candies are poured into a cup. I like the blue candies best. When I want candy, I turn over the dispenser and sort through the candies in the cup. I leave all of the blue ones in the cup and put the rest back. Then I turn over the hourglass a second time. The dispenser refills the cup so that I get exactly 20 candies, which I eat. I continue this way until the dispenser is empty.
How many blue candies will be in the final cupful?

Solved by: Paul Cleary, P.M.A. Hakeem, Mark Rickert, Kaustuv Sengupta

Cable (contributed by Don Dechman)
A flexible cable was hung across the Black Canyon between two points that were exactly 1 km apart and at the same elevation. During the cool night the cable length was calculated to contract by .2 meters. The cable dip was actually measured to decrease by .2 meters! What is the length of the cable after cooling?

Solved by: Mark Rickert, Brian Gladman, P.M.A. Hakeem, Paul Cleary, Naim Uygun, Kaustuv Sengupta

Cubic Equation (contributed by Denis Borris)
Find integers a, b and c such that
987654321a + 123456789b + c = (a + b + c)³
There are (at least) 3 solutions with a and b non-zero.

Solved by: Paul Cleary (doubly infinite family), Don Dechman (infinite family), Mark Rickert (infinite family), P.M.A. Hakeem, Andreas Abraham, Naim Uygun, Ionut-Zaharia Chirila, Arthur Vause (infinite family), Ender Aktulga (infinite family)

Old Foskins
       MegaTech Insurance Co. has just bought out Old Line Insurance from Stodgy Corp. They have put the 10,000 policies from Old Line into their computer system, and today they will begin accepting new policies. Everything is computerized, but just to make sure, they will follow Old Line's long-time practice and have Old Foskins double-check all the policies.
       The policies are stored alphabetically in the computer. Each morning a technician prints out the next 100 policies in sequence for Old Foskins to check by paper and pencil examination. Each evening 200 new policies are merged into the computer in alphabetic order.
       How long will it take Old Foskins to reach the end of the list?

Solved by: James Layland, Nick McGrath, Sudipta Das, Andreas Abraham, Bert Sheldon, Ankur Mathur, Mark Rickert, P.M.A. Hakeem, Paul Cleary

3 Families
Three families make a remarkable discovery. The sum of the ages of their members are all the same, the sum of the squares of the ages of their members are all the same, and the sum of the cubes of the ages of their members are all the same. Everyone in all 3 families has a different age, and nobody is more than 100 years old.
What is the smallest possible sum of their ages? Can this be done with 4 families?

Solved by: Don Dechman, Bimal Jit Kaur, P.M.A. Hakeem, Naim Uygun, Paul Cleary

Handshakes
On our Easy Math Puzzles webpage we have the following puzzle:

The Johnsons, the Jensens and the Jansons all hold a big party. Everyone shakes hands with every member of the other two families, 142 handshakes in all. Assuming that there at least as many Jensens as Johnsons, and at least as many Jansons as Jensens, how many of each family are present?

(A) Is there any number larger than 142 handshakes that still leads to a unique solution? (B) What is the largest number of handshakes if we are told that there is a different number of people in each family? (C) What is the largest number leading to a unique solution if we know that there are at least 2 people in each family? (D) What is the maximum if there are 4 families, each with at least 2 members, and each with a different number of members?

Solved by: Sudipta Das, Nick McGrath, Gaurav Agrawal, P.M.A. Hakeem

Tunnel (contributed by Denis Borris)
       Four safety engineers set out to inspect a newly cut tunnel through Mt. Popocaterpillar in the Andes. Each person walks at a different constant integer speed measured in meters per minute. In the tunnel there is a mine car which travels along a fixed track, automatically going from end to end at a fixed integer speed. When people board the car they may reverse its direction, but cannot change its speed.
       At noon on Monday all four engineers start at the south end, while the mine car starts at the north end. The first (fastest) engineer meets the car, and takes it some distance north. The engineer gets out and continues going north, while the car resumes heading south. Then the second engineer meets the car and also takes it some distance north. Likewise for the third and fourth engineers. All the people, and the mine car, travel continuously with no pauses. The inspectors always go north. Each person enters and exits the car at an integral number of minutes.
       All four engineers reach the end of the tunnel simultaneously. What is the earliest time this could happen?

Solved by: Paul Cleary, P.M.A. Hakeem, Antonio Ducay

The Census
       The census taker comes to the home of Mr. and Mrs. Lobotomy. He jots down the 4-digit house number and the ages of the wife and her somewhat older husband. Then he asks about their children.
       Mrs. Lobotomy tells him that they have 3 daughters and 3 sons, that the product of the 3 daughters' ages is the same as their house number, the sum of the 3 daughters' ages is the same as her own age, the product of the 3 sons' ages is also the same as their house number, and the sum of the 3 sons' ages is the same as her husband's age.
       The census taker is an accomplished mathematician, but after some time he determines that it is not possible to figure the ages of the sons or the daughters.
       Mr. Lobotomy then tells the census taker that the difference in age between the youngest daughter and youngest son is the same as the age of their cat. The census taker does not see a cat, and they have not mentioned its age, but he knows that it still would be impossible to determine the ages of the 6 children.
       The cat is 3 years old, and the sum of the ages of the oldest son and the oldest daughter is not 60. What are the ages of the 8 people in the Lobotomy family?

Solved by: Gilles Ravat, Sudipta Das, Rik Sheldon, Max H.B. Chan

Twelve Statements
Which of the statements in the box are true?

1	This is a numbered list of twelve statements.
2	Exactly 3 of the last 6 statements are true.
3	Exactly 2 of the even-numbered statements are true.
4	If statement 5 is true, then statements 6 and 7 are both true.
5	The 3 preceding statements are all false.
6	Exactly 4 of the odd-numbered statements are true.
7	Either statement 2 or 3 is true, but not both.
8	If statement 7 is true, then 5 and 6 are both true.
9	Exactly 3 of the first 6 statements are true.
10	The next two statements are both true.
11	Exactly 1 of statements 7, 8 and 9 are true.
12	Exactly 4 of the preceding statements are true.

Solved by: Praveen Yalagandula, Kathleen Coppersmith, Glenn C. Rhoads, Gilles Ravat, Colin Bown, Bappaditya Banerjee, Saurabh Agarwal, Sudipta Das, Steve Goldstein, Amitayu Pal, Janaki Mahalingam, Neeraj Parik, Gabriel Pirvan, Mark Thornquist, Jyotika Bahuguna, Ankur Mathur, Abhishek Daga, Mark Rickert, P.M.A. Hakeem, Kushal Khaitan, Kaustuv Sengupta

The Desert
Faye Rose Stume, the renowned Egyptologist, must drive her jeep across the Kanahara Desert and reach Khartoun, some 2500 miles away. The gas tank holds 15 gallons, and the jeep can carry up to 10 jerry cans of gasoline each holding 5 gallons. Gasoline can be poured from a can into the gas tank, or from one can into another without loss, but gasoline cannot be siphoned from the tank into a can. It is safe to stash cans of gas along the way and pick them up later. There is a large supply of gasoline and jerry cans in the town where she is starting, but they are both expensive. No gas or cans are available in the desert. The jeep gets 10 miles per gallon. How can she get there using the least amount of gasoline?

Solved by: Nenad Mihailovic, Don Dechman, P.M.A. Hakeem, Paul Cleary

Pirates (contributed by Sudipta Das)
       Aboard the ship 'Jolly Rogers' there are P pirates, numbered 1 to P, according to rank. A treasure chest containing N gold coins has been found. According to the rules of the pirate world, the highest ranking pirate decides how the treasure is to be split. Then all the pirates, including himself, vote on the decision. If a majority vote against the split then Pirate #1 is thrown overboard, and the next in rank takes over. The process is repeated until there is no majority voting against the way the split is decided.
       The votes of the pirates are governed by the following factors: A pirate values his life before money. A pirate will vote against any plan, unless that would cause him to be killed or to get less gold.
       How should Pirate #1 divide the treasure?

Solved by: Nenad Mihailovic, Nick McGrath, Tan Lye Huat, Gurpreet Singh Sawhney, Kushal Khaitan

Rubinian coins
The island of Rubinia issued 3 memorial gold coins between 1902 and 1999, inclusive. (All of their other coins are copper.) The value of each such coin is the same as the year when it was issued. For example, a coin issued in 1927 would be worth 27 rubins. Because of the unusual denominations, the Rubinians have devised a number of coin puzzles. Two of these puzzles are "What is the only way to make 535 rubins using 6 gold coins," and "Find the unique way to make 73 rubins with 8 gold coins."
Solve these two coin puzzles.

Solved by: James Layland, Keith Wood, Gilles Ravat, Sudipta Das, Dick Li, Nick McGrath, Amitayu Pal, Ritwik Chaudhuri, Bappaditya Banerjee, Amir Rubin, Rik Sheldon, Andreas Abraham, Janaki Mahalingam, Mark Thornquist, Mark Rickert, P.M.A. Hakeem, Pratheep Chellamuthu, Paul Cleary, Ionut-Zaharia Chirila, Ender Aktulga

Sliding Coins #2 (contributed by Rik Sheldon)
       Below are pictured 5 coins (represented by X and O). Your job is to get the 3 X's together and the 2 O's together by sliding the coins. You must slide two coins at a time, an X and an O. They must move together as a unit. You cannot move them further apart or closer together, and you cannot turn them around, that is turn XO into OX or OX into XO. You also cannot spread the 3 remaining coins apart or push them closer together. The coins must end up in a straight line with no gaps.
       It can be done in just 3 moves. Can you find them?

X  O  X  O  X

Solved by: Nick McGrath, Andreas Abraham, Janaki Mahalingam, Sujay Shastry, Balaji Gopalan, Arijit Bhattacharyya, Michael Mendelsohn, Bruce Langford, Mark Thornquist, Terje Kristensen, Ajay Agrawal, Nishanthi Sivanandanayagam, Jordan Canete, Inderpal Pandher, Denis Borris, Abhishek Daga, Pete Wiedman, Mark Rickert, Anupam Mittal, Rupa Kamat, Tan Lye Huat, P.M.A. Hakeem, Prateek Singal, Janice Wong, Kush Kumar, Jean-Charles Meyrignac (72 solutions)

Which Chair?
       Prof. Tai Rant gives out only one A+ each year in her course on Logic. This year, you and two other students got 100 on all the exams, and you REALLY want that A+. The professor says that she will give the 3 of you a test to determine the winner.
       There are 3 chairs in her office. After lunch each of you will sit in one of the chairs, and then she will paste two stamps on your forehead. She shows you the 4 white and 4 black stamps that she has. You will be able to see the stamps on the other two students' foreheads, but not your own, or the two leftover stamps. Then she will ask the student in the first chair to tell what color stamps are on his or her forehead. If the student cannot logically deduce the colors, she will move on to the second chair, then the third chair. If that does not decide the issue, she will continue around the circle of chairs until one of you gives the correct response, with correct reasoning, based on the stamps that are visible and the other students' answers.
       After lunch you are the first to arrive at the professor's office. Which chair do you choose?

Solved by: J. Hise, Glenn C. Rhoads, Sudipta Das, Mark Thornquist, P.M.A. Hakeem

A Sheet of Stamps
I have a sheet of stamps that I want to separate into individual stamps. How can I do it with the fewest possible tears?
Assume the stamps are all rectangular, all the same size, arranged in a uniform rectangular grid, each stamp is perforated along all 4 edges, and the sheet is very thin.

Solved by: Glenn C. Rhoads, Gilles Ravat, Janaki Mahalingam, Richard Farmbrough, Michael Mendelsohn, Mark Rickert

Bridge at Midnight (contributed by Denis Borris)
       Eleven people on a war games weekend come to a railroad trestle. It is midnight and there is no moon. Crossing is very dangerous because the ties are slippery and unevenly spaced, but they need to get over in the shortest possible time. The people are of widely differing ages and fitness. The time it will take the people to cross is 5, 10, 15, 20, 30, 30, 35, 45, 50, 55 and 65 minutes, respectively. Whenever 2 people cross together, they cross in the time of the slower person. Each person knows everyone's time.
       They have just 2 small flashlights that each cast only enough light to allow two people to cross safely. Once they begin crossing, both flashlights are in constant motion, being carried across and back on the bridge. They are never held waiting for someone to arrive. The flashlights can be handed off at either end, but not part way. Crossers never turn around or stop on the bridge. The last sets of crossers all finish together. Also, the two people who need 30 minutes hate each other and will not cross together.
       How long does it take for everyone to cross (A) if one person can carry both flashlights, (B) if one person cannot carry both flashlights? How is this accomplished?
       Please submit all answers in the form of a 5-column table or schedule. The left column should be the time, starting at T=0, the second column the people waiting to cross, the third column the people crossing the bridge, the fourth column the people returning, and the fifth column the people who have already crossed.

Solved by: Sandor Fuleki, Chris Schumann, Sudipta Das, S. Mohammed Rizwan, P.M.A. Hakeem

Mr. S and Ms. P
I have chosen two different numbers greater than N but less than M. I tell their sum to Mr. S and their product to Ms. P. The following conversation ensues:
Mr. S: I cannot determine the two numbers.
Ms. P: I cannot determine the two numbers either.
Mr. S: I still cannot determine the two numbers.
Ms. P: Now I can determine the two numbers.
Mr. S: Now I can determine the two numbers also.
Find the greatest value of M for which this puzzle has a unique solution, for N=1, N=2 and N=3.

Solved by: Sudipta Das

Huggy Bear
       The game of Huggy Bear is popular at large parties. The emcee gets all of the participants on the dance floor, and then calls out numbers between 2 and 12, inclusive. When a number is called, the players must form into groups of that number. Anyone left over is out. The game continues until there are only 2 people left. They are the winners.
       You have written a song for playing this game, which you are planning to record on a CD of party songs. In the song, at certain points, you will pause and call out a number. The players must then form groups of that size. These numbers will be the same each time the song is played.
       You would like your CD to be used with groups ranging from 20 up to N participants. You want N as large as possible, always leaving exactly 2 players at the end. How large can N be, and what sequence of numbers should you call? A short sequence is preferred.

Solved by: Nick McGrath, Sudipta Das, Richard Farmbrough, P.M.A. Hakeem

The Shipwreck (Contributed by Denis Borris)
       Five married couples are out for a pleasure cruise on a yacht when the boat hits a rock and sinks. They are left stranded on a small island 6 miles from shore. Luckily, there is a rowboat moored on the island large enough to hold 3 people, and the rowboat is equipped with 3 sets of oars and oarlocks.
       Each of the men can row the boat at 3 mph, and each of the women can row it at 1 mph. If two or more people row together, their speeds are added. That is, 2 men would row at 6 mph, a couple would row at 4 mph, etc.
       The husbands are all very jealous, and will not allow their wives to be with any of the other men unless they are also present, either in the boat, on the island or on shore. Several women may be together, but if any man is with them, all of their husbands must be there. (When the boat is loading or unloading, all of the people, both in and out of the boat, are considered to be together.)
       In the shipwreck, 7 of the people suffered neck injuries, and cannot turn their heads to see where the boat is heading, so one of the 3 uninjured people must be in the boat whenever it is being rowed.
       What is the least time required to get everybody safely to shore? (You can decide which people are injured.)

Solved by: Sandor Fuleki, Sudipta Das, Mark Rickert, P.M.A. Hakeem, Tan Lye Huat, Rohit Saraf

Heterodoxy
You have been hired to build the Temple of Heterodoxy for a fixed fee. The Temple must be rectangular, divided into two or more rectangular interior rooms, with each side an integral number of bozols. The outer dimensions and the dimensions of each room must all be different. For example, you could not have both a 5x9 room and a 9x11 room. The thickness of the walls is negligible.
What is the smallest possible area for the floor of the temple?

Solved by: Nick McGrath, P.M.A. Hakeem, Paul Cleary, Arthur Vause

Heterodoxy #2 (contributed by Jean-Charles Meyrignac)
You have been hired to build the Second Temple of Heterodoxy for a fixed fee. The Temple must be square with two or more square interior rooms whose sides are an integral number of bozols. There may be 2 rooms of any given size, but there cannot be 3 rooms all the same size.
What is the smallest possible area for the floor of the temple?

Solved by: Nick McGrath, Paul Cleary

The Manaslu Trail (Based on a concept from Denis Borris)
       Four monks from Tibet have crossed the border into Nepal, and are trying to reach safety in Dharapani by walking the 24 kilometers on the treacherous Manaslu Trail along the Dudh Khola. They are of differing ages and fitness, and walk at 8, 4, 3 and 2 kilometers per hour, respectively. When several people walk together, they walk at the speed of the slowest person.
       Along the way there are 5 hiding places, one every 4 kilometers. These are the only places that they can safely stop to rest and eat. Each hiding place is so small that only one person can occupy it at a time, but a monk can stay there as long as desired. For religious reasons, a monk can enter or leave one of these hiding places, or reach the end of the trail, only on the exact hour. It is unsafe for any monk to stop and wait anywhere other than inside the hiding places or at the ends of the trail.
       Because the trail passes through narrow gorges and dense forest it will be dark the entire time they are traveling, and because the trail is so narrow and steep they must use a light to keep from falling off. They have one large torch that two people walking together can use, and they have 9 candles that a single person can use. The torch is lit when they start, and will last 24 hours, but the candles last for only one hour each. The lama, the senior monk, who is the slowest walker, carries the torch and all of the candles. He will give the other monks only one candle at a time, and only at the start or at the hiding places.
       What is shortest possible combined transit time for the four monks?
       [Webmaster's note: in adapting this puzzle for the website I chose a real locale and background story. The actual Manaslu Trail in Nepal runs east from Dharapani, just north of the Dudh Khola (Brown River). It passes close to the Tibet border at points about 24 km east. Shortly before I posted the puzzle, the Panchen Lama escaped from Tibet using just such a trail, possibly the Manaslu Trail itself.]

Solved by: Pratheep Chellamuthu, P.M.A. Hakeem, Abhay Menon

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