The Contest Center - Harder Math Puzzles

Tougher mathematical puzzles for stronger solvers

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

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

Wrong! Wrong! Wrong! (contributed by Kaustuv Sengupta )
This long division is wrong. In fact, every digit is wrong.

				4	3	2	1
		-	-	-	-	-	-
1	5	)	9	5	8	6	7
			4	2
			-	-
				3	6	5
					1	2
				-	-	-
						7	2
						6	9
						-	-

Change all of the digits so that the division is correct. This is an exact division with no remainder. None of the numbers have leading zeroes.

Solved by: Naim Uygun, Ender Aktulga, Turgay Yüktaş

Seven Square Roots
Rationalize the denominator of the fraction
1
----------------------------------------------
√2 + √3 + √5 + √7 + √11 + √13 + √17

Celsius to Fahrenheit (contributed by Kaustuv Sengupta )
When the Celsius temperature 275 is converted to Fahrenheit, the result is 527. That is, the rightmost digit is moved to the leftmost position. Find the next Celsius temperature for which this is true.

Solved by: Ender Aktulga, Naim Uygun, Nick McGrath, Ionut-Zaharia Chirila, Turgay Yüktaş

Triple Cubic (contributed by Kaustuv Sengupta )
Find all positive real solutions x, y, z of the following system of equations:

x³ + y = 12x + 20
2y³ + z = 24y + 36
3z³ + x = 36z + 52

Solved by: Ionut-Zaharia Chirila, Ender Aktulga, Naim Uygun, Paul Cleary

1995 Sequence (contributed by Kaustuv Sengupta )
Let (T₀, T₁, T₂, ..., T₁₉₉₅) be a sequence of positive real numbers with T₀ = T₁₉₉₅, that satisfies T_i-1 + 2/T_i-1 = 2T_i + 1/T_i for all i=1, 2,...., 1995.
Determine the maximum value of T₀.

Solved by: Arthur Vause, Ender Aktulga, Ionut-Zaharia Chirila

A, B, C, D, E (contributed by Kaustuv Sengupta )
Determine all possible quintuplets (A,B,C,D,E) of positive integers, with A ≤ B ≤ C ≤ D ≤ E, that satisfy the equation

(A-1)*(B-2)*(C-3)*(D-4)*(E-5) = A+B+C+D+E.

Solved by: Naim Uygun, Paul Cleary, Ender Aktulga, Ionut-Zaharia Chirila, Turgay Yüktaş

Twelve Coins (contributed by Kaustuv Sengupta )
You have 12 coins that are indistinguishable, except for weight. Six of the coins weigh 24 grams, and six weigh 25 grams. Using a pan balance, how can you sort the coins using the fewest possible weighings? Within that constraint, what is the smallest possible expected number of weighings?

Solved by: Arthur Vause

Real Polynomials (contributed by Kaustuv Sengupta )
Find all polynomial functions F(x) that have real coefficients, and that satisfy F(x)F(2x²-1) = F(x²)F(2x-1).

Solved by: Arthur Vause, Ender Aktulga

Fifth Degree Diophantine (contributed by Kaustuv Sengupta )
Determine all integer solutions to the equation 5y⁴ + 2560y² = x⁵ - 65536. Solve the equation algebraically, without resorting to any computer program assisted technique.

Solved by: Ender Aktulga, Naim Uygun

Three Heirs (contributed by Kaustuv Sengupta )
       A wealthy man had three adult children all of whom were skilled at math and logic. To get a share of his inheritance each had to correctly determine a secret number which he had chosen. He told them that the number had four different non-zero decimal digits, in ascending order.
       He prepared three sealed envelopes each of which contained a clue. Envelope #1 contained the product of the four digits, envelope #2 contained the sum of the squares of the four digits, and envelope #3 contained the sum of the product of the first two digits and the product of the last two digits. For example, if the number were 1234, the first envelope would contain 1×2×3×4=24, the second envelope would contain 1²+2²+3²+4²=30, and the third envelope would contain 1×2+3×4=14. The envelopes were clearly marked. Each child was given one envelope. They each saw the number inside their own envelopes but could not see the numbers inside the other two envelopes.
       Then he explained the rules. After one hour each child could either submit a number or decline. Anyone who submitted a wrong answer would be eliminated and get nothing. If one or more submitted the correct answer they would receive equal shares of the inheritance. The contest would then end, with the others getting nothing. If no one submitted the correct answer they would be instructed to work on the problem for another hour. The process would repeat as often as necessary. Each of the children decided not to submit an answer unless they sure it was correct.
       The children were placed in different rooms where they could not communicate with one another. At the end of the first hour no one had submitted an answer. At the end of the second hour no one had submitted an answer. At the end of the third hour no one had submitted an answer. At the end of the fourth hour all three of them submitted the correct answer!
       What number had the father chosen?

Solved by: Arthur Vause, Ender Aktulga

Grant Distribution Among Districts (contributed by Shyam Sunder Gupta from his book Creative Puzzles to Ignite Your Mind (Springer https://lnkd.in/dmtqi273))
The head of state decided to distribute a grant among 25 Covid-affected districts, A1, A2, A3, A4, ..., A25, as follows: The first district, A1, gets 4% of the total grant. The second district, A2, gets 8% of the remaining grant. The third district, A3, gets 12% of the now remaining grant, and so on. Thus the n^th district gets 4n% of the available grant at its turn. The last district, A25, gets 100% of the last remaining grant.
Find the district that gets the largest share of the grant.

Solved by: Paul Cleary, Naim Uygun, Kaustuv Sengupta, Ender Aktulga, P.M.A. Hakeem, Nick McGrath, Ionut-Zaharia Chirila, Turgay Yüktaş

Grandma
       Three Silovian mathematicians, Adal, Biru and Cado meet at an actuarial convention. Adal says to the other two, "If you multiply your ages, the product is the year my mother was born." Biru says, "If you two multiply your ages, the product is the year my mother's mother was born." Cado says, "If you two multiply your ages, the product is the year my mother's mother's mother was born." All Silovians know that no Silovian woman has ever given birth younger than 15 nor older than 45.
       A fourth mathematician, Diba, arrives and says, "I heard your conversation, but I cannot determine any of your ages." Adal says, "We all have different ages." Diba says, "I still can't determine the age of any of you."
       What is the most recent year that this convention could have been held?

Solved by: Arthur Vause, Dave Mellinger, Ender Aktulga

How Tall?
One way to estimate the height of a tree is to stand a person of known height next to the tree. Hold your arm out towards the tree, and place your thumb and index finger so that the person is exactly framed between them. Then move your hand up so the thumb appears even with the top of the head, and note where on the tree the index finger appears. Continue this until the treetop appears bewtween the fingers. The estimated height is the number of times you sighted times the known height of the person.
Assume the person is 2 meters tall, the base of the tree is exactly level with your eyes, and you are standing 50 meters from the tree. How tall is the tree if your estimate is 20% too large? How tall is the tree if your estimate is 20% too small?

Solved by: Arthur Vause

Root Progression (contributed by Paul Cleary)
Find coefficients a, b, c, d and e so that the 10 roots of the polynomial equation x¹⁰ - ax⁸ + bx⁶ - cx⁴ + dx² - e = 0 are real values in arithmetic progression with a common difference of 4.

Solved by: Naim Uygun, Ender Aktulga, P.M.A. Hakeem, Ionut-Zaharia Chirila, Kaustuv Sengupta

2014 to 2019 (contributed by Amogh Keni)
Let p = 2014×2015×2018×2019 + 4. Without using a calculator, using only algebra, find [√p - (2017^2)].

Solved by: Paul Cleary, Ender Aktulga, Naim Uygun, Kaustuv Sengupta, Ionut-Zaharia Chirila

G-Series (contributed by Paul Cleary)
A G-series, or General Fibonacci Series, is defined as follows. Let a and b be any two positive integers. Set G₀ = a, G₁ = b and define the remaining terms recursively as
G_i+2 = G_i + G_i+1
The partial sums of the G-series are defined by P₀ = G₀ and
P_i+1 = P_i + G_i+1
Show that there are infinitely many pairs of positive integers i,j such that G_i evenly divides P_j.

Solved by: Arthur Vause, P.M.A. Hakeem, Ionut-Zaharia Chirila

Move 3 (contributed by Paul Cleary)
Find the smallest integer N such that when you move the last 3 decimal digits (in the same order) to the front the result is 2N.

Solved by: Arthur Vause, Mark Rickert, P.M.A. Hakeem, Claudio Baiocchi, Ender Aktulga, Naim Uygun, Ionut-Zaharia Chirila, Turgay Yüktaş, Kaustuv Sengupta

Decimal Dice
The DeciMate encryption scheme requires a sequence of true random decimal digits. The sequence will be hand-generated by throwing a set of dice to determine each digit. The dice are perfect cubes with 6 blank faces. Any number or symbol can be written on each face. Design such a set of dice so that each sum 0 through 9 has exactly the same probability.

Solved by: Nick McGrath, P.M.A. Hakeem, Arthur Vause, Doug Stryker, Mark Rickert, Paul Cleary, Kaustuv Sengupta, Ionut-Zaharia Chirila

Greatest Common Divisor (contributed by Paul Cleary)
When n=8, the expression ab(aⁿ-bⁿ) is divisible by 30 for all positive integers a and b, and 30 is the greatest such divisor. Find a positive integer n such that the greatest common divisor of ab(aⁿ-bⁿ) for all positive integers a and b is n.

Solved by: Arthur Vause

ABCD (contributed by Paul Cleary)
Without using a computer search, find positive values of a, b, c and d that simultaneously satisfy
       abc = 129960(a+b+c),
       abd = 108300(a+b+d),
       acd = 584820(a+c+d),
       bcd = 307800(b+c+d).

Solved by: Naim Uygun, Ender Aktulga, P.M.A. Hakeem

49 Cups (contributed by Paul Cleary)
49 empty cups are arranged in a 7×7 square. Each cup can hold up to 50 marbles. Every time you add marbles to or remove marbles from a cup you must add or remove the same number of marbles to or from the adjacent cups. What is the largest number of marbles you can have in the center cup when all of the other cups are empty?

Solved by: Arthur Vause, Claudio Baiocchi, Kaustuv Sengupta

Both Square (contributed by Paul Cleary)
Find all pairs of positive integers x and y such that x²+197y/2 and y²+197x/2 are both squares.

Solved by: P.M.A. Hakeem, Arthur Vause, Ender Aktulga

Square+27 (with Paul Cleary)
Find all positive integers x > y > z such that

x²+27y, y²+27z, z²+27x and x²+y²+z²+27

are all perfect squares.

Solved by: P.M.A. Hakeem, Naim Uygun, Ender Aktulga, Arthur Vause, Kaustuv Sengupta, Ionut-Zaharia Chirila

Four Squares (contributed by Paul Cleary)
Find all positive integers w > x > y > z such that 9w²+77x, 9x²+77y, 9y²+77z and 9z²+77w are all perfect squares.

Solved by: Arthur Vause, Ender Aktulga

Ten Squares (contributed by Paul Cleary)
Find ten positive integers a > b > c > d > e > f > g > h > i > j such that a²+5b, b²+5c, c²+5d, ..., j²+5a are all perfect squares.

Solved by: Arthur Vause, Ender Aktulga

Alternating Differences (contributed by Paul Cleary)
Let X₁, X₂, X₃, ..., X_n be a permutation of the integers 1,2,3,...,n. Consider the sum abs(X₁-X₃) + abs(X₂-X₄) + abs(X₃-X₅) + ... + abs(X_n-2-X_n). What is the mean value of this sum taken over all possible permutations?

Solved by: Arthur Vause, P.M.A. Hakeem, Ender Aktulga, Kaustuv Sengupta, Ionut-Zaharia Chirila

Integer Fractions #1
Find integers x,y,z > 1 such that the fractions

123x²+y

123x+y

45y²+z

45y+z

67z²+x

67z+x

are all integers.

Solved by: Paul Cleary (40 solutions), P.M.A. Hakeem (19 solutions), Naim Uygun, Arthur Vause, Ender Aktulga (4 solutions), Mark Rickert (25 solutions, plus 1 infinite family)

Integer Fractions #2
Find 3 distinct positive integers x, y, z such that the fractions

123x²+456y²+7890z²

123x+456y+7890z

456x²+7890y²+123z²

456x+7890y+123z

7890x²+123y²+456z²

7890x+123y+456z

are all integers. [** Extra credit if you find the smallest solution, in the sense that max(x,y,z) is smallest.]

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

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 its defenses.
       To defend our planet, we have constructed 30 deep space missile platforms, with a total supply of 400,000 missiles. The platforms are designed such that all of the missiles on a given platform must be launched simultaneously.
       Enemy ships can be detected with plenty of time for each missile to select and lock onto a different target and destroy it. The missiles are perfectly accurate, and each missile will destroy its target with absolute certainty. Any missile without an assigned target will harmlessly self-destruct. Missiles are expensive, so we want to expend as few as possible.
       When N₁ attacking ships are detected, the number of missiles launched will be determined by the "greedy" algorithm. First the platform with the most missiles ≤N₁ will be chosen. If N₂ more missiles are needed, the platform with the most missiles ≤N₂ will be chosen, and so forth. However, if this would not yield enough missiles, then the platform with the fewest missiles ≥N₁ will be chosen instead.
       We know that the enemy will send their ships in 1, 2 or 3 equal-sized waves. We need to be certain that the Earth will survive. We also want to waste as few missiles as possible. That is, the maximum number of missiles which self-destruct should be as low as possible. How many missiles should be on each platform?

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 at most 400,000 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 harmlessly 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, Kushal Khaitan

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 12 unit moves?

Solved by: Tim Joseph Clark, Turgay Yüktaş

Unit Cubes (contributed by Paul Cleary)
An A×B×C 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, Claudio Baiocchi, Arthur Vause, Ender Aktulga, Kaustuv Sengupta

Integer Solutions 173 (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, Arthur Vause, Ionut-Zaharia Chirila, Ender Aktulga, Naim Uygun

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, Arthur Vause, Doug Stryker, Claudio Baiocchi, Ender Aktulga

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, Arthur Vause

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, Ionut-Zaharia Chirila, Amogh Keni, Kushal Khaitan, Kaustuv Sengupta

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, Arthur Vause, Ender Aktulga

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 it is available. If it is already taken, then they are given a random empty room. What is the chance that the 101^st guest gets her reserved room?

Solved by: Mark Rickert, Gaurav Agrawal, P.M.A. Hakeem, Toby Gottfried, Nick McGrath, Paul Cleary, Doug Stryker, Ionut-Zaharia Chirila, Kushal Khaitan, Kaustuv Sengupta

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, Ionut-Zaharia Chirila, Kaustuv Sengupta

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, P.M.A. Hakeem, Kaustuv Sengupta, Ionut-Zaharia Chirila

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, Arthur Vause, Claudio Baiocchi, Ionut-Zaharia Chirila, Kaustuv Sengupta

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, Kaustuv Sengupta, Ionut-Zaharia Chirila

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, Mark Rickert

The 2N Sequence
It is possible to have sequences of positive integers that satisfy X_n > ∑(√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.

Part I solved by: P.M.A. Hakeem, Paul Cleary
Part II solved by: P.M.A. Hakeem, Paul Cleary

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, Kaustuv Sengupta, Ionut-Zaharia Chirila

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 were 3:2:1 and there were 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, Ender Aktulga

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, P.M.A. Hakeem, Arthur Vause, Ionut-Zaharia Chirila, Kaustuv Sengupta

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 the 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, Gregory Koch

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 an integer.

Solved by: Andreas Abraham, P.M.A. Hakeem, Ionut-Zaharia Chirila

Rankings (contributed by Sudipta Das)
The N² 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, Amogh Keni, Kaustuv Sengupta, Ionut-Zaharia Chirila

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, Ionut-Zaharia Chirila

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, Claudio Baiocchi, Ender Aktulga, Kaustuv Sengupta

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, Claudio Baiocchi, Ender Aktulga

1000 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, Arthur Vause, Gregory Koch, Claudio Baiocchi, Kushal Khaitan, Abhay Menon, Ender Aktulga, Kaustuv Sengupta, Ionut-Zaharia Chirila

Almost Magic Square
A Magic Square is an N×N 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 N×N 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 3×3 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, Paul Cleary

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