A billionaire, let's call him Gil Bates, decides to leave a portion of his fortune to his 7 children, and the rest to charity. He places the money for his children in a large chest.
When he dies, the first child says, "My siblings are all crooks; I'm the only honest one," and takes 1/7 of the money in the chest. Any part left over is given to the nanny. (That is, any part beyond the 7 equal shares.)
Then the second child does exactly the same thing, taking 1/7 of the remaining money, and giving any leftover portion to the nanny.
All 7 children do this, each taking 1/7 of the remaining money in the chest (even dollars, no change) and giving the balance to the nanny. The nanny gets a different amount from each child.
How much does the nanny get altogether?
Solved by: Nick McGrath, Sudipta Das, Andreas Abraham, Denis Borris, Arijit Bhattacharyya, Janaki Mahalingam, Aysha, N.S. Kamra, Swapna Munukoti, Mark Rickert, Paul Cleary, P.M.A. Hakeem
3 Bricklayers
A small company with 3 bricklayers is hired to build a barbecue. Each worker can lay a certain integral number of bricks a minute. The company could assign any combination of 1, 2 or all 3 workers to the job, and each combination would complete the barbecue in a different integral number of minutes. What is the smallest number of bricks in the barbecue?
Solved by: Nick McGrath, Mark Rickert, Paul Cleary, P.M.A. Hakeem
Election
Assume that 50,000,000 votes are cast in the next U.S. presidential election. What is the smallest number of these votes that a candidate could receive and still be elected president? [Your answer need not be probable, but it should have at least one chance in 1,000,000 of actually happening.]
Solved by: Carlos Rivera, Sen. Stephen Saland, Toby Gottfried, Mark Rickert
Trains
The railroad bridge across the Futile Mire has a northbound track and a southbound track. The tracks are parallel and exactly 2 miles long. Anyone attempting to cross the bridge on foot must stay between the rails on the chosen track. It is not possible to step outside the rails or to jump to the other track.
One day Tom and Sue decide to walk across the bridge, Tom walking north on the northbound track, and Sue walking south on the southbound track. When Tom and Sue are 1 mile apart they both hear trains whistles behind them. Trains are approaching on both tracks.
Tom and Sue can just escape by running to either end of the bridge. The athletic Sue can run twice as fast as the portly Tom. Both trains run at the same constant speed. How far apart are the trains?
Solved by: P.M.A. Hakeem, Dipin Singh, Paul Cleary
Landlord (contributed by Pete Wiedman)
A man has no money, but has a gold chain containing 23 links. His landlord agrees to accept one gold link per day in payment of the rent, and promises that he will not sell any of the gold until 23 days have passed.
The man hopes eventually to buy back the chain from the landlord so he wants to make as few cuts as possible. The renter also does not wish to overpay the landlord on any given day so it must be possible to make every number of links from 1 to 23 from the cut sections of chain. What is the smallest number of links that must be cut so that the man can make his payments as agreed?
Solved by: Mark Rickert, P.M.A. Hakeem, Paul Cleary, Claudio Meller
Red Hair (contributed by Douglas Tench)
Dan: How many children do you have, and what are their ages?
Nan: I have 3 children. The sum of their ages is 14. The product of their ages is equal to the number of the house across the street.
Dan: I need more information in order to determine your kids' ages.
Nan: My eldest child has red hair.
Dan: Thanks. Now I know their ages!
What are the childrens' ages?
Solved by: Andreas Abraham, Arijit Bhattacharyya, N.S. Kamra, Swapna Munukoti, Pete Wiedman, Ron Nelken, Mark Rickert, Paul Cleary, P.M.A. Hakeem
Clock Street
The house numbers on Clock Street are the same as the seconds on a digital clock, namely
00:00:00, 00:00:01, ..., 00:00:59, 00:01:00, ..., 00:59:59, 01:00:00, ..., 23:59:59
(A) The sum of the addresses up to and including Jack's house equals the sum of the addresses past Jack's house up to and including Jake's house. What are their addresses? (There are several solutions. Please give the largest.)
(B) The sum of the even addresses up to Jane's house equals the sum of the odd addresses past Jane's house up to and including June's house. What are their addresses? (There are several solutions. Please give the largest.)
Solved by: Andreas Abraham, Paul Cleary, P.M.A. Hakeem
Steps
There is a staircase with 10 steps. A child runs up the steps, either 1 step or 2 steps with each stride. How many different ways can the stairs be climbed?
Solved by: Ritwik Chaudhuri, P.M.A. Hakeem
Fair Toss
Suppose you have one fair coin, that is, a coin that comes up heads half the time and tails half the time. Show how to use this coin to choose fairly among N people.
Solved by: Nick McGrath, Andreas Abraham, Mark Rickert, Dwika Putra, P.M.A. Hakeem, Lev Bass, Ritwik Chaudhuri
Prisoners (contributed by Sudipta Das)
Attila the Hun lined up N prisoners in size places, from the shortest in the front to the tallest in the back, so that everyone could see the backs of the heads of everyone ahead, but nobody behind. He then announced the following logic game to decide their fates. "In my bag there are N white dots and N-1 black dots. I will place one dot on the back each of your heads. Then, starting with the last person in the line, I will ask each of you the color of your dot. If anybody gives a wrong answer you are all doomed. But, if even one of you gives me a correct answer, I will set all of you free. So, think carefully before saying anything." Attila then began the questioning procedure. Each prisoner responded "I do not know" until he got to the final prisoner, the shortest one, who correctly deduced the color of the dot.
What color dot was on each person's head? Assume that nobody took a guess, since the fate of all the prisoners depended on the answer.
Solved by: Nick McGrath, Andree Susanto, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem
Friday
What is the smallest and largest possible number of days from one Friday the Thirteenth to the next one?
Solved by: Carlos Rivera, Nick McGrath, Toby Gottfried, Andreas Abraham, Janaki Mahalingam, Jack Crawford, Arijit Bhattacharyya, Paul Cleary, P.M.A. Hakeem
XOXOXOXOX (based on an idea from Sudipta Das)
Two young math students are trying to develop a new notation for numbers for a class project. They want it to be based on completely different principles from decimal notation. Joe decides to try using a "word" of X's and O's, where the first letter tells you if the number is divisible by 2, the second letter indicates if it's divisible by 3, then 4, and so forth. An X means yes, and an O means no.
Here is how the first few numbers would be represented in Joe's system: 1=O, 2=X, 3=OX, 4=XOX, 5=OOOX, 6=XX.
Joe tells Amy that he has chosen a number between 1 and N, inclusive, and that he has represented it by a 9-letter word (indicating divisibility by 2 through 10). Amy, sitting across the table, looks at the word and says that the number must be A. Joe is puzzled because that is not the number he chose. Amy, seeing his confusion, quickly realizes what the problem is, and gives him the expected answer.
What is the largest possible value for N?
Solved by: Nick McGrath
Computer
A professor makes a long computer run every week. The run is divided into 100 parts. At the end of each part, the computer briefly displays how long that part took. After a few hours the professor checks the run's progress. When the current part finishes, the professor multiplies its time by 100 to estimate the total time for the run.
The professor finds that this estimated time is usually longer than the actual time. Why?
Solved by: Mark Rickert
The Party
Three people at a party discover that if you multiply the ages of any two of them, the product is the year in which the third person was born. In what year (since 1900) was the party held?
Solved by: Ritwik Chaudhuri, Nick McGrath, Sudipta Das, Amitayu Pal, G. Rajendri Prasad, Andreas Abraham, J-R Schrag, Taylor Freeman, Neeraj Parik, Rakesh Kumar Banka, S. Preethi Sudharsha, Ioana Scurt, Arijit Bhattacharyya, N.S. Kamra, Ed Leyro, Swapna Munukoti, Kiruba Manohari, Avinash Prakhya, Paul Cleary, P.M.A. Hakeem
3 Children
A woman has 3 children of different ages. She notices that if you reversed the digits in any of their ages (for example, 13 would become 31), they would become as much older than she is as she is now older than they are. How old is she?
Solved by: Ritwik Chaudhuri, Nick McGrath, Sudipta Das, Rakesh Kumar Banka, Taylor Freeman, Andreas Abraham, S. Preethi Sudharsha, Toby Gottfried, Nishanthi Sivanandanayagam, Ho Kyung Park, Ed Leyro, Kiruba Manohari, Jack Crawford, Avinash Prakhya, Ron Nelken, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Prateek Singal, Amitayu Pal
Red and Yellow (contributed by Nick McGrath)
A bag contains an unknown number of red balls and yellow balls. When N balls are drawn at random (without replacement) the probability that they are all yellow is 1/2. The number of balls in the bag is the minimum for this to happen.
If the first N balls were all yellow, what is the probability that the next ball drawn is red? [Express the probability as a function of N.]
Solved by: Sudipta Das, Toby Gottfried, Andreas Abraham, Douglas Tench, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem
Sum of Digits (contributed by Sudipta Das)
(A) Find all squares such that the sum of any 2 digits is also a square. (B) Find all cubes such that the sum of any 3 digits is also a cube.
Solved by: Andreas Abraham, P.M.A. Hakeem, Ritwik Chaudhuri
Complementary Numbers
Two whole numbers are called complementary if their corresponding digits add to 9. For example, 397 and 602 are complementary
3 9 7
6 0 2
-----
9 9 9 Two squares that are complementary are 0 and 9. Two primes that are complementary are 2 and 7. Show that there are no others. Solved by: Sudipta Das, Andreas Abraham, Arijit Bhattacharyya, P.M.A. Hakeem
Doors
There is a special place in the netherworld reserved for mathematicians who play with prime numbers. It consists of a corridor 1000 miles long lined with 1,000,000 doors. Each of the 6666 souls imprisoned there is assigned a prime number, 2, 3, 5, 7, 11, etc.
At the start of the first month all of the doors are closed. Each person must walk down down the corridor, opening or shutting every P-th door, according to the given prime. For example, the soul assigned #5 opens or shuts doors 5, 10, 15, 20, etc.
At the end of a month they have all reached the end of the corridor, and they must start back, performing the operation in the opposite order. For example the #7 soul opens or closes doors 999994, 999987, 999980, 999973, and so forth.
How many months will it be before the doors are all closed again?
Solved by: Sudipta Das, Nick McGrath, Ritwik Chaudhuri, Andreas Abraham, S. Preethi Sudharsha, Mark Rickert, Janaki Sivaramakrishnan, P.M.A. Hakeem, Paul Cleary
Dartboard
I decided I wanted one of those dartboard puzzles for this website, the kind where you are given the scores for each ring of the dartboard, and you have to determine how many darts fell in each ring to get a given total.
I chose 4 different scores for the 4 rings, then I picked a total N that could only be reached one way. It required 10 darts. I noticed that each of the totals N+1, N+2, N+3, ..., N+10 also could be reached only one way, and in each case fewer than 10 darts were needed.
What is the smallest value of N for which this is possible?
Solved by: Nick McGrath, Andreas Abraham, Mark Rickert
Dartboard #2 (contributed by Mark Rickert)
After I chose the total N, in the puzzle above, it also turned out that there was only one possible set of scores for the 4 rings that met all of the conditions. What is the smallest value of N for which this can happen?
Tour
Lance Foodlemyer won this year's Tour de Hoboken with a remarkable ride. He started along Mugger Alley, up Junkyard Hill, through Slum Lane, down into Pothole Valley, rode a level circle around the Gasworks and Refinery Park, then returned along the same route, finishing in 8 hours flat. He averaged 20 mph on the level, 15 mph on the uphills, and 30 mph on the downhills. What was the total length of the trip?
Solved by: Sudipta Das, Chris Schumann, Lev Bass, Felix Pirvan, Nick McGrath, William Alber, Douglas Tench, Andreas Abraham, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem
Handshakes
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?
Solved by: Gilles Ravat, Chris Schumann, Ken Duisenberg, Colin Bown, Sudipta Das, Lev Bass, Nick McGrath, Andreas Abraham, Chad Mossing, S. Preethi Sudharsha, S. Mohammed Rizwan, N.S. Kamra, Kiruba Manohari, Ron Nelken, P.M.A. Hakeem
Afflictia
On the populous island of Afflictia, 20% of the people have apopsia, 30% have kinomatosis, and 40% have spurge. What percentage have none of these?
Solved by: Sudipta Das, Nick McGrath, William Alber, Marina Plaksina, Andreas Abraham, Toby Gottfried, Ioana Scurt, Pete Wiedman, Mark Rickert, Dwika Putra, P.M.A. Hakeem, Geoffrey Pitman
The History Final
On Dr. Drydust's history final you could get any integer score from 0 to 100, inclusive. Fay got 3/4 as much as Jan. Pat got 14 points more than Jan. Lee got 2/3 as high as Pat. Dale got the average of all their scores.
What score did each student get?
Solved by: James Layland, Padmini Gopalakrishnan, J. Hise, K. Ramanathan, Keith Matlak, Jim Thompson, Alan Sears, Neeraj Parik, Steve Walzer, Wayne Lucas, Michael Dufour, Gilles Ravat, Gopalan Harish, Chris Schumann, Ed Leyro, Gaurav Jain, J. Michael Anderson, Nitin Agrawal, Sudipta Das, Ritwik Chaudhuri, Nick McGrath, Hai He, Amitayu Pal, Mike Bullock, William Alber, Hareendra Yalamanchili, Andreas Abraham, Ron Nelken, Janaki Mahalingam, N.S. Kamra, Janaki Sivaramakrishnan, Paul Cleary, P.M.A. Hakeem
Two Solids
I have solid models of a triangular pyramid and a square pyramid, with a square base and 4 triangular sides. In both models all of the edges are 1 foot long. If I place the models together, with a triangular face of each coinciding, how many faces does the resulting figure have?
Solved by: Don Doyle, Colin Bown, Lev Bass, J-R Schrag, Andreas Abraham, Sudipta Das, P.M.A. Hakeem
Three Accountants
Effie Foodlemyer needs to hire an accountant. Three people apply for the job. She decides to use a mathematical puzzle to test each accountant's numeric skills. She chooses 4 whole numbers, and tells the first applicant their product. The accountant immediately tells her the 4 numbers. Too easy. She tells the second applicant the sum of the squares of the 4 numbers. Again, the accountant tells her the 4 numbers immediately. Still too easy. So she tells the last applicant the sum of the 4 numbers. That accountant is unable to tell her the 4 numbers. [Note: she interviews the 3 accountants separately.]
What 4 numbers did she choose?
Solved by: Nick McGrath, Andreas Abraham, Denis Borris, P.M.A. Hakeem
Auction (contributed by Lucas Jones)
A real estate speculator is going to a condemned property auction. The speculator places a large sum of cash into 15 packets in a way that allows for any purchase amount up to $30,000. Each packet contains the smallest possible number of bills, for example there could not be 2 fives in place of a ten.
At the auction the speculator buys an abandoned gas station for $8,480, giving the cashier packets 6, 9 and 14, together containing precisely the required amount.
How many bills were in these 3 packets?
Solved by: Colin Bown, Nick McGrath, Sudipta Das, Andreas Abraham, Ron Nelken, N.S. Kamra, Toby Gottfried, Avinash Prakhya, Paul Cleary, P.M.A. Hakeem
Cube
Let N be any integer greater than 1. Show that there are positive integers A, B, C and D such that A! B! = C! D! and A + B = N^3. (Here A!, called the factorial of A, is the product of all the positive integers up to and including A, so that 4!=1×2×3×4=24.) For example, if N=2 then N^3=8, and we find 3! 5! = 1! 6! and 3 + 5 = 8.
Solved by: Nick McGrath, Ritwik Chaudhuri, Andreas Abraham, Mark Rickert, Paul Cleary, P.M.A. Hakeem
Tilting Tower
In a large level rectangular field a tall wooden tower was built for training parachute jumpers. The tower is in the shape of a truncated rectangular pyramid. That is, the base is a rectangle, and the top is a smaller rectangle parallel to the base. The longer sides of the field, the base of the tower and the top of the tower are all oriented east to west.
After several years of use, a strong gale tilted the tower towards the west, so that the western edge of the top was higher than the eastern edge. (The top was still a rectangle with its eastern and western edges parallel to the eastern and western edges of the field.) The engineers determined that the tower was still fit for use, but to prevent further tilting they stretched cables tightly from the corners of the field to the nearest corners of the top of the tower. The lengths of these cables, going clockwise are 95, 109, 125, and X meters. What is X?
Solved by: Gilles Ravat, Don Doyle, Colin Bown, Nick McGrath, Andreas Abraham, Janaki Mahalingam, Mark Rickert, P.M.A. Hakeem
Swimming Pool
An architect is designing a kidney-shaped swimming pool for a millionaire. The pool will be 120 feet long overall, with the diameter of the larger circular end 50 feet, and the smaller circular end 36 feet. Around the pool there will be a 4-foot wide marble walkway, with a thin tight-fitting rubber edging on either side. The length of the edging on the inside (next to the pool) is 380 feet. What is the length of the edging on the outer side of the walkway?
Solved by: Gilles Ravat, Don Doyle, Colin Bown, Matt Batsa, Nick McGrath, Paul Cleary
Pythagoras
Every high school math student knows that 32+42=52. Prove this by cutting up the first two squares into the smallest number of pieces and reassembling them into the third square.
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Solved by: Nick McGrath, Sudipta Das, Toby Gottfried, Chad Mossing, N.S. Kamra, Ho Kyung Park, Pete Wiedman, P.M.A. Hakeem, Paul Cleary
Square Roots (contributed by Ritwik Chaudhuri)
Let R(x) represent the square root of x rounded to the nearest integer. Thus R(12) would be 3 and R(13) would be 4. Without using a computer or calculator, and without calculating any square roots, evaluate
1/R(1) + 1/R(2) +1/R(3) + ... + 1/R(2550)
Solved by: Sudipta Das, Nick McGrath, Hareendra Yalamanchili, Amitayu Pal, Andreas Abraham, S. Preethi Sudharsha, Arijit Bhattacharyya, Avinash Prakhya, Paul Cleary, P.M.A. Hakeem, David Urman
Triangular Numbers
A triangular number is the sum of any of the series 1+2+3+...+n. The first five triangular numbers are 1, 3, 6, 10 and 15. Show that there are an infinite number of triangular numbers that are also squares.
Solved by: Carlos Rivera, Stephane Higueret, Sudipta Das, Ritwik Chaudhuri, Nick McGrath, Hareendra Yalamanchili, Arijit Bhattacharyya, Paul Cleary, P.M.A. Hakeem, Mark Rickert
Plug
A room has a 20-foot ceiling. From the ceiling hang two electrical cords 16 feet and 18 feet long. At the end of one is a plug, and at the end of the other is a receptacle. The electrician is tall, and can reach up to 8 feet from the floor. How far apart can the two cords be, yet still be possible for the electrician to connect them?
Solved by: Don Doyle, Gilles Ravat, Ken Duisenberg, Chris Schumann, Colin Bown, Stephane Higueret, Lev Bass, Ely Levy, Nick McGrath, Amitayu Pal, Mike Bullock, Sudipta Das, Hareendra Yalamanchili, Rakesh Kumar Banka, Chad Mossing, Taylor Freeman, Andreas Abraham, N.S. Kamra, Jean-Charles Desjardins, Arijit Bhattacharyya, Paul Cleary
Size Places
Without using a calculator or logarithms, arrange the following expressions in order from smallest to largest
(A) 10^10^10 (B) 3^3^3^3 (C) 2^2^2^2^2
The operator ^ indicates exponentiation, so 2^5 means 2 to the fifth power, which is 32. Note that a^b^c means a^(b^c) and not (a^b)^c, so that 2^3^2 is 512, not 64.
Solved by: Ritwik Chaudhuri, Hareendra Yalamanchili, Arijit Bhattacharyya, P.M.A. Hakeem
Prime Triangle
Show that there cannot be a triangle in which the 3 sides and the area are all primes.
Solved by: Carlos Rivera, Ely Levy, Sudipta Das, Nick McGrath, Le My An, Andreas Abraham, Andree Susanto, Arijit Bhattacharyya, P.M.A. Hakeem, Ritwik Chaudhuri, Paul Cleary
Any 14
Suppose you choose 14 of the integers from 1 to 1111, inclusive. Show that in any such set there are two subsets having the same sum for their elements.
Solved by: Carlos Rivera, Ely Levy, Nick McGrath, Hareendra Yalamanchili, Ritwik Chaudhuri, Andreas Abraham, P.M.A. Hakeem, Mark Rickert
Sin
Find all angles A and B such that sin(A+B) = sin(A)+sin(B).
Solved by: Sudipta Das, Ritwik Chaudhuri, Hareendra Yalamanchili, P.M.A. Hakeem
Reciprocals #1
Find all positive integers A, B and C such that 1/A + 1/B = 1/C.
Solved by: Ritwik Chaudhuri, Nick McGrath, Andreas Abraham, N.S. Kamra, Jesse Wiedman, P.M.A. Hakeem
Reciprocals #2
Find all integers A, B and C such that 1/A + 1/B + 1/C = 1.
Solved by: Ritwik Chaudhuri, Nick McGrath, Sudipta Das, Douglas Tench, Andreas Abraham, P.M.A. Hakeem, Paul Cleary
Reciprocals #3 (contributed by Ritwik Chaudhuri)
Find all integers A, B and C such that (1 + 1/A) (1+ 1/B) (1 + 1/C) = 3.
Solved by: Sudipta Das, Nick McGrath, Anirban Bhattacharyya, Amitayu Pal, Andreas Abraham, P.M.A. Hakeem, Paul Cleary
Two of Four
There are many sets of 4 distinct integers a < b < c < d where the sum of any 2 is a square. Find the set for which d is smallest.
Solved by: Carlos Rivera, Sudipta Das, Nick McGrath, Andreas Abraham, Mark Rickert, P.M.A. Hakeem, Paul Cleary
2001 (contributed by Ritwik Chaudhuri)
If x + 1/x = -1 find the value of x^2001 + 1/x^2001
Solved by: Nick McGrath, Sudipta Das, Hareendra Yalamanchili, Andreas Abraham, Alex Liu, Andree Susanto, Amit K. Sahu, N.S. Kamra, Arijit Bhattacharyya, Diptajit Bhattacharyya, Venkat Ramanan Nambirajan, Kaustuv Sengupta, P.M.A. Hakeem, Paul Cleary
Sequence of Squares
(A) Show that there is an infinite sequence of distinct positive integers a, b, c, d, ... for which ab+1, bc+1, cd+1, ... are all squares.
(B) Show that there is an infinite sequence of distinct positive integers a, b, c, d, ... for which ab+1, abc+1, abcd+1, ... are all squares.
Solved by: Colin Bown, Carlos Rivera, Nick McGrath, Ritwik Chaudhuri, Andreas Abraham, Douglas Tench, Arijit Bhattacharyya, Mark Rickert, P.M.A. Hakeem
Five Clocks
I have 5 clocks. One clock is accurate, one clock gains a minute every hour, one clock loses a minute every hour, one runs at double speed, and the last one runs backwards. Today at 4:12 PM the five clocks all agreed on the time. When will that happen next?
Solved by: Chris Schumann, Sudipta Das, Ritwik Chaudhuri, Nick McGrath, Mike Bullock, Andreas Abraham, S. Preethi Sudharsha, Arijit Bhattacharyya, Paul Cleary, P.M.A. Hakeem
Equations
Find the real roots of the following set of equations:
| x3yz + x2zy2 + 3z2xy = 315 2y2zx2 - 4yzx3 + xz2y = 49 6z2yx - 4zx2y2 + 9zx3y = 6 |
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