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.

Missile Platforms #1 (with Paul Cleary)
       The planet Zpft has launched a fleet of spaceships to destroy the Earth. We know that Zpft has a total of 300,000 attack ships, but we don't know how many they sent. Earth will be destroyed if even one ship gets through.
       To defend our planet, we will construct 300,000 missiles, which will be launched from 30 deep space platforms. The missiles are perfectly accurate, and each missile will destroy its target with absolute certainty. They are also expensive, so we will launch only one missile for each attacking ship, if possible.
       The platforms are designed such that all of the missiles on a given platform must be launched simultaneously. Enemy ships can be detected at a distance of 100,000,000 miles, which gives plenty of time for each missile to select and lock onto a different target and destroy it.
       You anticipate that the enemy may send their ships in 2 waves, or possibly even 3 waves. You want to maximize the probability that Earth survives. How many missiles should be on each platform?
       To make this manageable, assume that 1, 2 or 3 waves are equally likely. Consider two separate cases, (A) each wave will have the same number of ships, or (B) for a given number of waves, every possible combination of ships is equally probable. For example, for 2 waves 100 and 25 is just as likely as 53,726 and 371. You can get credit for solving either or both cases.

Missile Platforms #2 (with Paul Cleary)
You are Lord Imperator of the Zpftian Space Command. You have learned the Earth's defense plan. You don't know how many missiles will be on each platform, but you do know that there will be 300,000 missiles on 30 missile platforms, and that all of the missiles on any platform will be launched simultaneously. You know that each missile will choose a different target, and will follow it until it is destroyed. You also know that if a missile cannot find a target, it will self-destruct.
You decide to send your attack fleet in several waves, each 200,000,000 miles apart. You know that the Earthlings have planned for this. On each subsequent wave they will launch the fewest missiles possible to destroy all of the ships they can see. What is the fewest ships you can send, yet be certain of destroying the Earth?

Solved by: Nick McGrath

Cubic Lattice (contributed by Paul Cleary)
In the cubic lattice formed by the integer points (a,b,c) in 3-space, what is the probability of returning to the origin after making 2N unit moves?

Unit Cubes (contributed by Paul Cleary)
An AxBxC rectangular block has been divided into ABC unit cubes. Some of these cubes have faces on the surface of the block, while others are entirely interior. Show that for any positive integer N there is a block with N times as many interior cubes as surface cubes. Find the dimensions of such a block having the smallest volume.

Solved by: Nick McGrath, P.M.A. Hakeem

Integer Solutions (contributed by Paul Cleary)
Find all integers N for which the expression

√(N+70710678√173) - √(N-70710678√173)

has an integer value.

Solved by: Nick McGrath, P.M.A. Hakeem

The Codemaker
I want to make a simple code by writing each of the letters from A to Z one or more times into a square grid, in some order unknown to my opponent. I will represent each letter by two letters, using any two letters that bracket the chosen letter in the grid. For example, to represent H in the following grid I could use any of the pairs DP, EN, SM, GI, IG, MS, NE or PD. For O the choices would be CD, DC, RM or MR.
I want to be certain that no letter pair can represent two different letters. The grid below would not be acceptable because MY could represent either T or Z. Also, any given letter should not be bracketed by the same pair more than once, and the grid cannot contain any blank cells. What is the largest valid grid?

C	O	D	E	S	A
B	F	G	H	I	J
K	L	M	N	P	Q
R	T	U	V	W	X
Y	Z	M	A	K	E
R	C	A	R	O	M

Solved by: P.M.A. Hakeem

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, Paul Cleary

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. No guessing is allowed.
       (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

Best-of-Five
In a best-of-five tennis match Pete Shamprowess led André Allgassy the whole way during the first two sets. André won more games in the first set than in the second. Pete won the first game in the third set. Pete won the match, even though André won 25% more games.
What were the scores in each of the 5 sets?

Solved by: James Layland, Morgan Pembroke, Nathan Johnson, Jim Thompson, Neeraj Parik, Gilles Ravat, Gopalan Harish, Colin Bown, Sudipta Das, Hai He, Nick McGrath, Ritwik Chaudhuri, Hareendra Yalamanchili, Saurabh Agarwal, Rik Sheldon, Janaki Mahalingam, Taylor Freeman, Roshan Chutkey, Harit Baveja, Ajay Agrawal, Amit K. Sahu, Sudip Roy, Avinash Parwaney, Ankur Mathur, Avinash Prakhya, Rohit Choudhry, Mark Rickert, Derrick Niederman, P.M.A. Hakeem, Prateek Singal, Suresh Purohit, Ray Wolfe

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, Toby Gottfried, Nick McGrath, Paul Cleary

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. 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, Toby Gottfried, Ray Wolfe

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, P.M.A. Hakeem

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

4 Dice (contributed by Paul Cleary)
I have a set of 4 fair dice (the probabilities of each outcome are equal). The faces of these dice are marked 1 through 6, 1 through 7, 1 through 8, and 1 through 9.
Find a set of 2 dice which have the same probabilities as my set of 4 dice. For example, my dice have a probability of 1 in 3024 of throwing 4, and 1 in 756 of throwing 29. Your dice must have the same. Ideally, your two dice should be as closely matched as possible, that is they should have the same number of faces marked 3, the same number marked 4, and so forth, to the greatest possible extent.

Solved by: P.M.A. Hakeem

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, Mark Rickert

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 must 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, and that all links in the new chain must be welded, even those at the ends. 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.
       Part II. The same situation as above, but you wish to keep the maximum number of welds as small as possible.

Solved by: P.M.A. Hakeem

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, Ritwik Chaudhuri, P.M.A. Hakeem

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, P.M.A. Hakeem

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, P.M.A. Hakeem, Mark Rickert

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, P.M.A. Hakeem

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, P.M.A. Hakeem

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, Ritwik Chaudhuri

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 / (²√2 + ³√3 + ⁵√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, Pratheep Chellamuthu.

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 positive integers to make the rows, columns and diagonals all 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

Interactive Online Games
<<< PREV	HOME	MAP	NEXT >>>