The Contest Center - Harder Math Puzzles

Tougher mathematical puzzles for better solvers

W	I	N
I		O
N	O	W

Harder Puzzles
The Contest Center
59 DeGarmo Hills Road
Wappingers Falls, NY 12590

W	I	N
I		O
N	O	W

Like this site?
Please tell a friend

Please sign our
Guestbook

We will post the names of anyone who solves
any of these puzzles.

Mixer
A bowl is full to the rim with 1 liter of salt water at 10% concentration. A liter of pure water is slowly poured into the bowl. (Assume no splashing and instantaneous mixing.) What percentage of salt will be left?

Solved by: P.M.A. Hakeem, Gaurav Agrawal

3 Hats
       The wizard has placed hats on the heads of three logicians. He explains that he has written a positive integer on each hat, and that one of the numbers is the sum of the other two numbers. Each person can see only the numbers on the other people's hats.
       The wizard offers a prize to the first person who can logically determine the number. The wizard starts questioning the logicians in order, starting over again if nobody can state the number with certainty.
       (1) Prove that someone will always determine the number. (2) Prove that will always be the person with the highest number. (3) Find an explicit formula for the number of rounds needed until the number is found.

Solved by: P.M.A. Hakeem

Permutation
The order of a permutation is the number of times that permutation must be repeated until the objects return to their original positions. For example the permutation (2,3,1) on the letters ABC successively generates BCA, CAB and ABC, so its order is 3.
What is the highest possible order of a permutation on a list of 100 elements?

Solved by: Mark Rickert, Gaurav Agrawal, P.M.A. Hakeem

Balls and Walls
Imagine a frictionless plane with short walls located at some of the integer points. The walls are tilted 45° to the horizontal x-axis, so that a ball moving horizontally will bounce off in a vertical direction (with no loss of energy), and vice-versa.
A ball located at the origin is shot horizontally in the +x direction. Show that either (1) the ball will escape, getting arbitrarily far from the origin, or (2) the ball will eventually return to the origin passing through in the +x direction. (In other words, show that the ball cannot get trapped elsewhere, and cannot return to the origin going in the opposite direction.)

Solved by: Gaurav Agrawal

Hotel (contributed by Pete Wiedman)
The Fort del Mar Hotel reserves all its 101 rooms for travelers and each guest is assigned a room in advance. The first guest arrives but has forgotten his room number. The hotel clerk, who does not have access to the reservations book, randomly puts him in one of the rooms. As the rest of the guests arrive they are given their reserved room if available or if already taken, are given a random empty room. What is the chance that the 101st guest gets her reserved room?

Solved by: Mark Rickert, Gaurav Agrawal, P.M.A. Hakeem

Chests
You are in a room with 2N Plaminians and C locked chests. One of the chests contains a treasure. Half of the Plaminians are consecrated, and know which chest contains the treasure. The other Plaminians do not know which chest has the treasure. All of the Plaminians know which ones are consecrated, but you do not know which Plaminians are consecrated, or which chest has the treasure.
The Plaminians have agreed that you may ask each one a yes/no question, and they will answer truthfully if they know the answer, and randomly if they do not know. What sequence of questions will give you the greatest chance of locating the chest with the treasure? For what values of N and C is success guaranteed?

Solved by: Nick McGrath, Gaurav Agrawal

Consecutive Integers (based on an idea from Denis Borris)
Consider sequences of consecutive integers
   A, A+1, A+2, A+3, ..., B-1
   B, B+1, B+2, B+3, ..., C-1
   C, C+1, C+2, C+3, ..., D-1
Show that if A and B are chosen so that the sum of the first sequence is a square, then you can find C, D, E, etc., so the sum of each subsequent sequence is also a square.

Solved by: Nick McGrath, Andras Erszegi, Mark Rickert, P.M.A. Hakeem

Bug on a Cube (contributed by Gaurav Agrawal)
A bug is placed at one corner of a wire frame in the shape of a cube. At the diagonally opposite corner is a piece of sugar. The bug crawls along the 12 wires of the frame searching for the sugar. At each of the 8 corners the bug randomly chooses one of the 3 wires to follow next (including the one it just traveled). What is the expected number of edges the bug will traverse until it reaches the sugar? What if the bug never doubles back on the wire it just crossed?

Solved by: Nick McGrath, Andreas Abraham, Bob Dillon, Mark Rickert

Phone Squad
Foodletown has a Volunteer Emergency Response Team which assists the police and fire departments. The members are alerted by phone. The police or fire chief will notify the captain of the V.E.R.T., who will notify the members. The captain will call certain members, who may call other members, and so forth. Each caller gives the location and nature of the emergency, and also may give up to 2 pieces of information to direct the phoning process, such as "I have called Smith, but I have not called Jones," or "You call Smith next and I will call Jones next." Assume that each call lasts exactly 1 minute.
How should the calling be organized such that all members are notified within the shortest possible time? You need to address two cases. In the ideal case, every member is reached, and the calling proceeds by a fixed plan. In the real-world case, there is a 50% chance that any given member is reached. (The captain is always reached.) In both cases, give the expected number of minutes if the squad has 100 members.

Rational
If a and b are rational numbers, and p and q are positive integers which are not squares, prove that a√p+b√q is either 0 or irrational.

Solved by: Nick McGrath, Arijit Bhattacharyya, Ritwik Chaudhuri, Paul Budney, Gaurav Agrawal

Flipper (contributed by Nick McGrath)
The game Flipper is played with 2 coins which may have any integer denomination from 1 to 99, inclusive. The coins are tossed, and the value of the toss is the sum of the denominations of those coins that show heads. The higher sum wins. In case of a tie, the game is replayed.
Two people play this game. One person has coins valued 1 and X, the other person has 5 and X. It is found that the game is fair, that is, the two people win in exact proportion to the total value of their coins. What is the value of X?

Solved by: Gaurav Agrawal, Andreas Abraham, Douglas Tench, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem, Tan Lye Huat

Flipper #2 (contributed by Nick McGrath)
Four people play the game Flipper described above. They decide to play on an elimination basis, namely the two winners of the first round games play each other in the second to establish an overall winner. Lots are drawn to decide who plays who in the first round.
Each player has a coin of value X, and each has a second coin valued A, B, C, and D, respectively. (a) If X > A > B > C > D, what is each player's probability of being the overall winner? (b) If the probability of each player being the overall winner is proportional the value of the two coins, and if A is prime, what are the values of X, A, B, C and D?

Solved by: Gaurav Agrawal, Andreas Abraham, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem

Flipper #3 (contributed by Nick McGrath)
Two people play 3 games of Flipper, as described above. In the first game they use coins valued A,X and B,X. In the second game they use coins valued A,Y and B,Y. In the third game they use coins valued A,Z and B,Z. All 3 games are fair, so that the probability of each player winning is proportional to the value of the coins.
A is odd, A > B and X > Y > Z. Find A, B, X, Y and Z with smallest A.

Solved by: Gaurav Agrawal, Andreas Abraham, Mark Rickert, P.M.A. Hakeem

Dice
       A die, in various gambling games, is a cube with its 6 faces numbered 1 through 6. However, in other games the 6 faces may have other numbers. If A and B are dice whose faces may have any whole numbers, we say that A beats B if the probability that A shows a higher number than B is greater than the probability that B shows a higher number than A. In other words, of the 36 equally probable outcomes, A shows a higher number than B more often than B shows a higher number than A. (For example, if A has 1,2,2,2,3,4 and B has 1,1,2,3,3,3 then of the 36 outcomes, A is higher 15 times, B is higher 13 times, and they tie 8 times. So A beats B.)
       It has been known since the 1950's that it is possible to have 3 dice, A, B and C such that A beats B, B beats C, and C beats A. Several such sets of 6-sided dice have appeared in the literature. However, as all Dungeons and Dragons enthusiasts know, you can have dice with any number of sides. Many D&D players have sets of dice with anywhere from 4 to 24 or more faces.
       Find the minimal set of such dice. If the numbers of faces are a, b and c, then the sum of the squares a²+b²+c² should be minimal, and likewise the sum of the squares of all the numbers on all the faces should be minimal.

Solved by: Nick McGrath, Mark Rickert

Dice #2
Let a die A beat a die B as defined in the previous problem. Find a set of 5 dice such that each die beats 2 of the others. In other words, all 5 of the dice are median. As before, use the fewest sides and smallest markings possible.

Solved by: Mark Rickert

The 2N Sequence
It is possible to have sequences of positive integers that satisfy X_n > Sum (sqrt(X_i) for i=1,2,3,...,2n). There is one such sequence which is smallest. Find the first 5 terms, and the 100th term of this sequence.

Solved by: Nick McGrath

Blacksmith
       A blacksmith has forged a chain of N links. He is told that this is the wrong length, and that he should cut the chain into pieces that can later be reassembled into a chain of the required length. (The cut links can be rewelded to join the various pieces, or used by themselves.) The correct length is not known yet, but it will be between A and B, inclusive.
       When the required length is determined the new chain will be needed as quickly as possible, so all of the cutting should be done in advance. The lengths of the pieces should be chosen so that the expected number of welds is as small as possible, assuming all lengths from A to B are equally likely. Find the lengths of the pieces for the following cases:
(1) N=1000, A=1, B=1000.
(2) N=1000, A=100, B=900.
(3) N=1000, A=200, B=800.
       Solve the same 3 cases when the maximum number of welds must be minimized.

Euler's Year (contributed by Nick McGrath)
The great mathematician Leonhard Euler proved at least one new theorem every day. In order not to overtax himself, however, he proved no more than 50 theorems in any month. Prove that there is a sequence of consecutive days in a year where Euler proves exactly 125 theorems.

Solved by: Gaurav Agrawal

Secret
       Two people have found that they must communicate important secret messages with each other. However, they have made no arrangements in advance. Their only channel for communicating is by public telephone lines that can be assumed to be passively bugged. That is, enemies may be listening to and recording all of the messages in both directions, including all discussions they may have concerning how the secret messages may be concealed, but these enemies cannot alter the messages, insert their own messages, or prevent any messages from getting through.
       Both of the people know some mathematics, but neither one knows much about cryptography, and certainly nothing that was developed after World War I. They both have older model personal computers with modems, and both have a moderate degree of programming skill. The enemy, however, can be assumed to have massive, unlimited computing power and expertise. The messages are vital, but not urgent, so they can take several hours per message, if necessary.
       How can they arrange to send their messages securely?

Rich Uncle (contributed by Sudipta Das)
A rich uncle of Jack, John and Jim gave them some valuable coins in the ratio A:B:C respectively, where C<=B<=A<10. For example, if the ratio is 3:2:1 and there is a total of 36 coins, then Jack would get 18, John would get 12, and Jim would get 6. At first, he distributed the coins randomly among the three, but the ratio was not A:B:C. Jack returned a number of coins equal to the percentage that he was supposed to receive. So (using the same ratios) if he got 30 coins, he returned 15 (50%). John and Jim returned coins in a similar fashion. Uncle then divided all the returned coins equally among the three. Believe it or not, the coins each had after this ended up exactly A:B:C.

Tom: And that, Dick, is what happened. Uncle gave out the minimum number of coins possible for that ratio, and for this strange method of distribution.
Dick: How many coins did Uncle give out?
Tom: This many (shows him a number).
Dick: And how many did each nephew receive?
Tom: Figure it out yourself.
Dick: Ok, I need hints.
Tom: Only one of them received that number over here (points to a day in his calendar).

And before even seeing the number, Dick shouts "GOT IT!" How many coins did each receive?

Solved by: Nick McGrath, Denis Borris

The Mummy (contributed by Sudipta Das)
       Explorers Tom, Dick and Harry have managed to reanimate the mummy of the ancient Egyptian mathematician Fuht-al-Mahir. The Mummy is very grateful, and promises to give them diamonds. Each of the explorers must pick a number from 1 to N (the Mummy's lucky number), and the Mummy will give them the diamonds in those proportions.
       The Mummy explains that initially he will give them the diamonds in an entirely different ratio, which he will determine mathematically. Then he will take a portion of each explorer's diamonds back, in the ratios the 3 men have chosen. Finally, he will divide the diamonds he took away evenly among the 3 explorers, ending up with their chosen ratios. [For example, if N were 3 and the 3 explorers chose the ratio 3:3:1 the Mummy would give them 7, 7 and 0 diamonds initially, take back 3, 3 and 0 diamonds (that is, 3/7 of 7 diamonds, and 1/7 of 0 diamonds), leaving them with 4, 4 and 0, then give them each 2 of the 6 diamonds he took. They would then have 6, 6 and 2 diamonds, which is in the chosen 3:3:1 ratio.]
       The Mummy warns them that if this is not possible, because initially one of them must receive a negative or fractional number of diamonds, then they will get nothing. Otherwise, he will give them the smallest number of diamonds that allows him to distribute them as he indicated, without splitting any diamonds.
       What ratio should the explorers request in order to get the most total diamonds, and how many diamonds will they get?

Solved by: Nick McGrath

Three Accountants #2
Jeffie Foodlemyer needs to hire an accountant. Three people apply for the job. He decides to use a mathematical puzzle to test each accountant's numeric skills. He chooses 4 integers greater than 1, and tells the first applicant their product. The accountant immediately tells him the 4 numbers. Too easy. He tells the second applicant the sum of the squares of the 4 numbers. Again, the accountant tells him the 4 numbers immediately. Still too easy. So he tells the last applicant the product of the 3 smallest numbers plus twice the square of the largest number. That accountant is unable to tell him the 4 numbers. [Note: he interviews the 3 accountants separately.]
What 4 numbers did he choose?

Solved by: Nick McGrath, Sudipta Das, Andreas Abraham, P.M.A. Hakeem

Repunits (contributed by Nick McGrath)
A repunit is an integer consisting entirely of ones, such as 1, 11, 111, etc. Let N be any integer not divisible by 2 or 5. Show that there is an integer M such that the product MN is a repunit.

Solved by: Steve Spindler, Douglas Tench, Arijit Bhattacharyya

Fibonacci Squares (contributed by Nick McGrath)
Fibonacci numbers are integers in the series 1, 1, 2, 3, 5, 8, 13, ... where each additional term is the sum of two preceding terms.
Find all integer solutions to the following equations:
(1) A²+B²+C²=D² where A, B and C are Fibonacci numbers, and D is any integer.
(2) A²+B²+C²+D²=E² where A, B, C, D and E are Fibonacci numbers.
(3) A²+B²+C²+D²+E²=F² where A, B, C, D, E and F are Fibonacci numbers.

Solved by: Denis Borris, Andreas Abraham

Fibonacci Squares #2
Fibonacci numbers are integers in the series 1, 1, 2, 3, 5, 8, 13, ... where each additional term is the sum of two preceding terms.
Prove that there are an infinite number of solutions to A²+B²+C²+D² = E² where A, B, C and D are distinct Fibonacci numbers, and E is any integer.

Solved by: Andreas Abraham

Rankings (contributed by Sudipta Das)
The N^2 participants in a contest are seated in an N by N square (N rows of N seats each). All the participants have been assigned different ranks, according to their performance. Each person asks his/her immediate neighbors (sitting left, right, ahead or behind) what rank they have. A participant will be happy if at most 1 neighbor has a better rank. What is the maximum possible number of happy participants?

Solved by: P.M.A. Hakeem

Pete and Jimmy (contributed by Nick McGrath)
Old Jimmy and young Pete are both tennis champions. They have played each other many times, each winning exactly half of the matches. When both are fresh Jimmy is much the better player, but he tires rapidly so his probability of winning the k^th set is p^k, where p is the probability that he wins the first set.
If they always play best-of-five matches, what is the value of p?

Solved by: Martin Rubin, Gaurav Agrawal, Andreas Abraham, Mark Rickert, P.M.A. Hakeem

Tick Tock
How many times in 12 hours can the tips of the hour, minute and second hands of a clock form an equilateral triangle? If the hour hand (the shortest hand) is 6 cm, what must the lengths of the other hands be?

Solved by: Mark Rickert

Rationalize
Rationalize the denominator of the fraction
1 / (sqroot(2) + cuberoot(3) + fifthroot(5))

Solved by: R. Robinson Rowe

An Average Student
Finley Fogg is a student at Drudgery High. After every test he figures his cumulative average, which he always rounds to the nearest whole percent. (So 85.49 would round down to 85, but 85.50 would round up to 86.) Today he had two tests. First he got 75 in French, which dropped his average by 1 point. Then he got 83 in History, which lowered his average another 2 points.
What is his average now?

Solved by: Neeraj Parik, Michael Dufour, Gilles Ravat, Chris Schumann, Sudipta Das, Nick McGrath, Ritwik Chaudhuri, Andreas Abraham, Toby Gottfried, Mark Rickert, P.M.A. Hakeem

Filling a Box
Show that it is possible to fill a rectangular box with rectangular blocks so that (1) the dimensions of all of the blocks are integers, (2) no edge of a block coincides with an entire edge of the box, and (3) no two blocks have identical faces (that is, you could not have blocks of size A×B×C and A×B×D). Find the box of least volume for which this is possible.

Solved by: Nick McGrath

Maximus (contributed by Sudipta Das)
       Maximus is one of 3 or more prisoners of war who have been lined up before the Roman Emperor. The Emperor orders the execution of the prisoners in the following manner:

1) The prisoners would be numbered from first in the line to last as 1, 2, 3, 4, ..., N.
2) The first and last are immediately executed.
3) If there are still 2 or more prisoners remaining, prisoners 2, 3, ... (N+1)/2 are executed.
4) If there are still 3 or more prisoners remaining, start again from Step 1.

       For example, if there were 5 prisoners to start then prisoners 1 and 5 would be executed in Step 2, and prisoners 2 and 3 would be executed in Step 3, leaving prisoner 4 alive.
       Is there a strategy by which Maximus can save himself? What if Maximus also wants to save his friend Romulus?

Solved by: Nick McGrath, Ritwik Chaudhuri, P.M.A. Hakeem, Gaurav Agrawal

Doors (contributed by Anirban Bhattacharyya)
Suppose there are 1000 doors namely D1,D2,D3,...,D1000 and there are 1000 persons namely P1,P2,P3,...,P1000. First P1 opens all the doors. Then P2 closes all even numbered doors. P3 changes every third door (i.e., he closes D3, opens D6 and so on). Similarly Pm changes every m^th door. Finally P1000 opens D1000 if it were closed, and closes it if it were open.
At the end how many doors remain open?

Solved by: Sudipta Das, Nick McGrath, Carlos Rivera, Ritwik Chaudhuri, Ole Jann, Andreas Abraham, Gaurav Agrawal, S. Preethi Sudharsha, Purushottam Dixit, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem, Paul Cleary

Almost Magic Square
A Magic Square is an NxN array of distinct integers (preferably the consecutive integers from 1 to N²) where each of the rows, columns and 2 main diagonals have the same sum. Now, imagine an NxN grid where one box has been blacked out. Is it possible to fill in distinct integers to make the rows, columns and diagonals have the same sum?
The diagrams illustrate the best attempts for the 3x3 case. All of the rows and columns have the same sum. In the first grid one of the diagonals has the same sum; the second grid is magic; in the third grid the 4 corners add to the magic sum. What about larger grids?

4	8
2	3	7
6	1	5

7		5
2	4	6
3	8	1

3	7	2
8		4
1	5	6

Solved by: Mark Rickert

Send us an email to submit answers to these puzzles, to ask for help, or to submit new puzzles.
Be sure to change the $ to an @ in our email address.

Quick Links

Tournament of Riches		One on One
<<< PREV	HOME	MAP	NEXT >>>