Chris Long's monthly mathematical puzzle column for Central New Jersey Mensa's Forvm.
Think & Derive is a column that I write for the Central New Jersey Mensa publication Forvm. Problem #1 in each column will range in difficulty from easy to moderate, problem #2 from difficult to impossible. The answer to each puzzle will be given in the next column.
Waldo is one of the leaders in the Cucumberland scouting organization and he's trying to help two scouts, Titus and Una, earn the coveted merit badge Wilderness Survival with Cheese. They want to make camp, but first need to cross a river to do this. They only have one canoe, which can hold up to 165 pounds without sinking. Waldo weighs 150 pounds, while Titus and Una both weigh 75 pounds. They have with them 50 pounds of gear (sleeping bags, crackers, napkins), a 25 pound wedge of cheddar, and a 20 pound wedge of mozzarella. What is the fewest number of trips necessary to transport everyone and everything, counting each way across the river as one trip? Note that Cucumberland tradition dictates that they may neither break down the gear nor eat any cheese until they make camp.
Waldo is having a party to celebrate Titus and Una's merit badge success. There are a total of 12k guests at this party (where k is a positive integer), including Waldo, Titus, and Una. Each guest knows exactly 3k+6 people at the party, and people are assumed to not know themselves (Cucumberlanders are not renowned for introspectiveness). Knowing is mutual; if A knows B, then B knows A. For any two given guests, a certain number of people at the party know both. The remarkable thing is that this number is the same regardless of who the two guests are! How many guests are at the party?
This was based on a problem in a recent Iranian mathematics competition.
Waldo is trapped in an underground dungeon which has seven doors to choose from. One door hides a secret escape route from the dungeon, and the rest conceal a passage that transports poor Waldo right back to where he started, and so disoriented that he's unable to recall which doors he previously chose. How many doors do you expect that Waldo will have to open before he escapes from this trap?
Waldo and Basil are doing a bit a gambling by flipping a fair coin repeatedly. If the sequence head-tail comes up first, Basil gives Waldo a dollar, tail-head first, Waldo gives Basil a dollar. For example, if the first 2 flips are tail-head, Basil wins a dollar; if the first 3 flips are head-head-tail Waldo wins a dollar; if they're tail-tail-tail no one wins a dollar (although if the next flip comes up heads, Basil wins a dollar). As soon as anyone wins everything is reset, i.e. head-tail-head is a win for Waldo followed by a head, not a win for Waldo followed by a win for Basil. After a good while they're still about even, and so to make things more interesting Basil suggests that they should instead each choose a sequence of length five. In fact, Basil offers to give Waldo $1.50 to $1 odds if Waldo will take the sequence head-head-head-tail-tail, i.e. if this sequence comes up before Basil's sequence, Waldo wins $1.50; if Basil's sequence comes up first, he only wins a $1.
Waldo reasons that since the number of heads and tails in this sequence are as equal as possible, he should have at least a 50% chance of winning no matter which sequence Basil chooses, and so he has the better deal. Is Waldo right?
I'm thinking of a word that means someone who collects cheese labels. What is it?
This puzzle was seen floating around the internet.
Spike has a large bag of candies, each of which is one of 5 possible different flavors: apple, banana, cherry, dutch chocolate, and elderberry. Assuming he can fit up to 10 candies in his mouth at once, how many different flavors can he make? Note that 1 apple and 1 banana is the same flavor as 2 apples and 2 bananas (just a larger amount), but that 1 apple and 2 bananas is not the same as 2 apples and 1 banana. Also note that he has more than 10 candies of each flavor.
For a gold star, generalize the number of flavors and how many can be eaten at one time.
Last month we heard about some strange critters that live on Cucumber Island, lasamanders. There are some even stranger critters that live in Cucumberland proper, macheleons. As with the lasamanders there are three different types: azure, blue, and cream. When two macheleons of a different type touch, they will both take on the color of the third type, e.g. azure touching blue results in two cream macheleons. If there are 1994 azure, 1995 blue, and 1996 cream macheleons, is it possible for them to all become the same color by an appropriate series of meetings? For example, if there are 2 azure and 2 blue macheleons to start with, this is possible by an azure touching a blue, and then the remaining azure touching the remaining blue, leaving us with 4 cream macheleons.
For a gold star, give necessary and sufficient conditions on the starting numbers of each type of macheleon so that it's possible for them to all become the same color. For another gold star, if it's possible, what can be said about this color?
My first three and last three are the same, and many people stay with me when they die. Who am I?
You're looking for an 11-letter word.
Molly is playing the videogame Space Squeakers, and she finally loses her last turn after several hours (and after killing numerous invading space mice). She was rather surprised to discover that she had scored the maximum number of points she could have while averaging exactly 9975 points per turn. If in Space Squeakers you start with 3 turns and you earn an extra turn with every 10000 points you score (e.g. you earn an extra turn at 10000, another at 20000, another at 30000 etc.), what was Molly's final score?
Optional: For a gold star, describe the set of scores Molly could have achieved while still averaging 9975 points per turn.
On Cucumber Island there are some strange critters, called lasamanders. There are three different types: alloy, bronze, and chrome. When two lasamanders of a different type meet, one will eat the other and take on the color of the third type, e.g. alloy meeting bronze results in a single chrome lasamander. If there are 1994 alloy, 1995 bronze, and 1996 chrome lasamanders, is it possible for them to reduce to a single lasamander by an appropriate series of meetings? For example, if there are 2 alloy and 1 bronze lasamanders to start with, this is possible by an alloy meeting the bronze and becoming chrome, then the remaining alloy meeting the chrome and becoming bronze.
Optional: For a gold star, give necessary and sufficient conditions on the starting numbers of each type of lasamander so that it's possible for them to reduce to a single lasamander. For another gold star, if it's possible, what can be said about the color of the last lasamander?
Aquatic creature cycles briefly to arise.
For those unfamiliar with the type of puzzle, it's of the form "This word, when you do this to it, becomes this other word". You're looking for a 7-letter word, and I'll take either the before or after.
Spike is taking a series of exams, and it turns out that he'll have to score a 97 on the last one in order to average 90 for the entire series. But even if he scores as low as a 73, he'll still average an 87. How many exams were in the series?
Waldo wants to draw an equilateral triangle (all sides the same length), and wants to do it as neatly as possible. To that end, he's using a large piece of graph paper made up of squares, and will have each edge go between two points on the graph paper.
Can such a neat triangle be drawn? In other words, does there exist an equilateral triangle in the real plane such that each vertex has the form (m,n) with m and n both integers?
"Irish wood is the best," Tom SAID.
I'm going to experiment with one off-beat verbal puzzle per column. In the problem above, you have to find a 6-letter word to replace SAID with that makes the above a nice Tom Swifty. A Tom Swifty is a phrase in which what Tom states is punningly related to how he says it; some examples:
"I love sugar in my coffee," Tom said sweetly.
"I'm dying," Tom croaked.
Donald Knuth, one of the most famous computer scientists in the world (and who was first published as a kid in Mad Magazine) believes that it's possible to make any positive integer by starting with a single 3 and then using some combination of the operations of factorial !, square-root sqrt(), and greatest integer []. Note that n! = 1*2*...*n (e.g. 6!=720), and that [x] is the greatest integer less than or equal to x (e.g. [3.14]=3). As an example, we can make 26 by [sqrt((3!)!)], since 3!=6, 6!=720, sqrt(720)=26.8, and [26.8]=26.
Show that it's possible to make 10.
Here's one from the USA Mathematical Talent Search; it's not easy, but it's certainly doable.
For a positive integer n, let P(n) be the product of the nonzero base 10 digits of n. Call n "prodigitious" if P(n) divides n. What is the maximum number of consecutive prodigititious positive integers n?
Hints: the answer is not 12; meditate on numbers ending in a 3, 6, or 9
From Charles Trigg's excellent collection Mathematical Quickies:
During a certain period of days in Cucumberland recently it was observed that when it rained in the afternoon, it had been clear in the morning, and when it rained in the morning, it was clear in the afternoon. It rained on 100 days, and was clear on 19 afternoons and 95 mornings. How many days were there altogether?
Another from Mathematical Quickies:
Spike has made up a polynomial with integral coefficients that has his age as a root, i.e. x stands for his age at his last birthday. Spike gives it to Molly as a puzzle; she tries x=7 and finds the polynomial gives 77. She then tries a larger number for his age and gets 85. Spike realizes that Molly is just having some fun with him, since she knows he's older than the two ages she's tried. He tells her to hurry up, and is rather surprised when she immediately tells him how old he is without any more trials.
How old is Spike?
A classic from Aaron Friedland:
The Gashouse Gorillas and the Mudville Mudders have each played the same number of games so far this season. The Gorillas have a .664 average, and the Mudders have won 70 games. Which team is ahead? Note that each team plays 162 games in a season, and that the team's average is the number of games won divided by the number of games played rounded off to three decimal places.
Another classic from Hugh ApSimon:
Waldo and Basil are washing windows as part of their spring cleaning regimen. They're working on Waldo's house, which is on Pi Street (a horizontal road with parallel vertical walls on either side). Waldo set his ladder with its foot at the base of the south wall, leaning squarely against the north wall; Basil set his ladder with its foot at the base of the north wall, leaning squarely against the south wall. The two ladders just touched each other.
Waldo's ladder is 3 feet longer than Basil's ladder, and reaches 4 feet higher up the north wall than Basil's does up the south wall. The point at which the two ladders touch each other is 5 feet 10 inches above the ground. How wide is Pi Street?
The three of us made some bets:
Who am I?
Given that I*MENSA = ZZZZZZ and that each different letter stands for a different digit, what's Z?
Consider the sum:
ABC + DEF + GHI = JJJ
If different letter represent different digits, and there are no leading zeros, what does J represent?
No instructions. When you get the right answer, you'll know it.
HARD+HOKY=LOLY
HOKY+CIA=PYKYL
Mortimer, Waldo, Basil, Spike and Molly all live in separate houses in Cucumberland. I happened to come across them in the town square recently, and tried to get the current house number of each to keep my little black book current. None would tell me outright what number they lived at, but:
Using the fact that the houses in Cucumberland are numbered from 1 to 50, what number did each person live at?
Which planet is usually closet to the planet Pluto? For simplicity, assume that the orbits are concentric circles in a plane and that each planet has a different constant angular velocity, and so for a given position of any planet, each of the other planets has an equal probability of being at any point on its orbit.
Waldo mentioned the problem they had getting to the Cucumberland fair recently (see the Cheese and Fleas puzzle), and Mortimer recalled a similar occurence from the his days as a boy. It seems that three couples staying on an island wanted to cross the water using a boat that could only hold two people at at time. In those days, it was considered improper for a woman to be with a man who was not her husband unless her husband was also present. How many trips were required? Each way counts as one trip.
For each positive integer n, let A_n be the number of digits in the binary representation of n, and let B_n be the number of ones in the binary representation of n. What is (1/2)^(A_1+B_1) + (1/2)^(A_2+B_2) + (1/2)^(A_3+B_3) + ... ?
This is much easier than it looks.
Waldo is a bit absentminded, and the only way he can remember his own phone number is that if you divide it by its reverse, you get an integer greater than one. What's Waldo's phone number?
Rameses wishes to build a great pyramid for his internment. The structure will have a square base and be solidly composed of cubical stone blocks. Each level of the pyramid contains one less block per side as the pyramid rises. Rameses has available an initial work force of 35,000 slaves. Each morning the available labor pool is divided into work crews of 17 slaves each. Any remainder that cannot form a full crew gets the day off but are available the following day. Each crew can lay one block of the pyramid each day. Unfortunately, the heat of the desert sun causes the death of one member of each crew each day. Work ceases on the project when it can be determined that there will be insufficient slaves available to raise the pyramid one more level. Each stone block measures 3 meters per side.
How many days will it take to construct Rameses' pyramid? How tall will it be? How many of the original slaves survive the construction?
This classic was printed in the October 1992 issue of PuzzleSIGns, the Mensa puzzle SIG publication. I don't know the original source.
Waldo, Basil, Molly, two wedges of cheddar, two wedges of mozzarella, and Rufus the dog are going to go to the annual Cucumberland cheese competition and dog show. Waldo's little sports car can only seat two objects at a time; Waldo, Basil and Molly can all drive the car. If Waldo is not around, Basil will eat the cheddar; if Basil is not around, Waldo will eat the mozzarella. If Molly is not around, Rufus will eat the cheese and bite Waldo and Basil. Can they all get to Cucumberland without anything bad happening? If so, what's the smallest number of trips needed (each way counts as one trip)?
Waldo is having a party and has 50 guests, among whom is his brother Basil. Basil starts a rumor about Waldo; a person hearing this rumor for the first time will then tell another person chosen uniformly at random the rumor, with the exceptions that no one will tell the rumor to Waldo or to the person they heard it from. If a person who already knows the rumor hears it again, they will not tell it again. What's the probability that everyone, except Waldo, will hear the rumor before it stops propagating?
A fair coin is flipped and then hidden. Molly has an 80% chance of guessing the state of the coin (heads or tails), Spike has a 70% chance, while poor Waldo only has a 10% chance of a correct guess. Basil wants to devise a scheme using all of these guesses to predict the state of the coin. What level of accuracy does the best scheme give?
You are given a bag of white and black stones, where you know the number of each type of stone in the bag. Reach in and draw two stones. If they are the same color, keep them out and put a black stone in the bag. If they are different colors, keep the black one out and put a white stone back in the bag. Repeat until only one stone is left.
What can be said about the color of the last stone?
The Cucumberland grocery has six cheese wedges of different sizes, weighing 15, 16, 18, 19, 20, and 31 pounds. Five wedges are cheddar and only one wedge is mozzarella. Waldo bought two wedges of cheddar, and Basil also bought cheddar, but twice as much by weight as Waldo. How much does the mozzarella wedge weigh?
Waldo and Basil were recently invited to a party attended by four other pairs of siblings, for a total of ten people. During the party various handshakes took place, but no person shook their own hand or the hand of their sibling. At the end of the party Waldo asked each person, including Basil, how many different people they shook hands with, and was surprised to note that every number was different! How many hands did Basil shake?
Prince Waldo went to fight a 3-headed, 3-tailed dragon. He has a magic sword that can, in one stroke, chop off either one head, two heads, one tail, or two tails. This dragon is of a type related to the hydra; if one head is chopped off, a new head grows. In place of one tail, two new tails grow; in place of two tails, one new head grows; if two heads are chopped off, nothing grows. What is the smallest number of strokes required to chop off all the dragon's heads and tails, thus killing it?
After graduating from college, you have taken an important managing position in the prestigious financial firm of Waldo & Spike. You are responsible for all the decisions concerning takeover bids, and your immediate concern is whether to try a takeover of Basil's Brokerage. There is no doubt that you will be successful if you are the first to bid and that this will be profitable for the firm and you in the long run. However, you know that there exist another n financial firms, similar to Waldo & Spike, that are also considering the possibility. Although you are likely to be the first one to move, you know that just after a takeover there is a lot of adjustment that needs to be done. In fact, for a period of time following any takeover the successful firm becomes a prime candidate for a takeover which will cost the job of whoever is responsible for takeovers (in other words, you). Among all financial firms it is common knowledge that the managers responsible for takeovers are rational and intelligent. What is your best response?
We were sitting around recently discussing birthdays (as we often do), and Mortimer mentioned that once on his father's birthday in 1937 someone (who didn't know it was his birthday) asked his father when he was born. He told him that he turned x years old in the year x^2, and that if you added his current age to the number of the month he was born in, it equaled the day of the month on which he was born. When was Mortimer's father born?
Eight players participated in the recent Cucumberland chess tournament; each player played all of the others exactly once. The winner of a game received 1 point and a loser 0; draws are allowed, giving each player 1/2 point. Now, it turned out that everyone received a different number of points. Furthermore, Molly, who came in second, earned as many points as the four bottom finishers put together. What was the result of the game between Waldo, who came in third, and Basil, who came in seventh?
Mortimer recently had another birthday. When someone mentioned that he was getting up there in years, he replied that he was actually quite young. Indeed, he pointed out, he is the youngest age such that the sum of the divisors of his age, not including the age itself, exceeded his age, yet the sum of no subset of these divisors equaled his age. How old had Mortimer just turned?
Define the sequence f(n) in the following manner:
f(0) = 0, f(n) = n - f(f(n-1))
What is f(1000000000000)?
I was sitting around with my friend Waldo and his grandfather Mortimer last week, and the topic of birthday surprises came up. Mortimer mentioned that one of the greatest surprises that he has had involved his grandfather, who happens to have had the same birthday that Mortimer has. One year the family was celebrating this double birthday, and during the events Mortimer proudly mentioned to his grandfather that not only he had just turned as old as the last two digits of the year he was born in, but he was also a prime number of years old, and each of the two digits making up his age was also a prime. Mortimer was floored when the older man thought for a second, turned to him, and said that the same thing had just happened to him! What year did this occur, and how old had Mortimer and his grandfather just turned?
I was sitting around with my friend Waldo, his nephew Spike, and Spike's friend Molly recently. I happened to have two tickets to a new movie in my pocket that I had just purchased, and I mentioned this and noted that there were two four-digit numbers on the tickets and that the sum of all 8 digits was 25. Waldo asked if any digit appeared more than twice out of the 8, which I answered, and then Spike asked if the sum of the digits of either ticket was equal to 13, which I answered also. Much to my surprise Molly immediately told me what the two numbers were! What were they?
When Mortimer turned x years old he noticed to his surprise that between them x^2 and x^3 included all the digits from 0 to 9, with none repeated. How old was he?
When Waldo recently did a conversion of a positive integral Celsius temperature c=275 to its Fahrenheit equivalent f (which turned out to be 527), he noticed to his amazement that he could have simply moved the last digit of c to the front to obtain f. Doing some intense calculations he failed to discover the next largest such example. Does one exist, and if so, what is it?
Mortimer and his grandson Waldo have the same birthday. For six consecutive birthdays Mortimer is an integral number of times as old as Waldo. How old is each at the sixth of these birthdays?
The Intel clock-doubled 486DX2-66 CPU chip operates by executing a certain fraction x of instructions totally on chip at a doubled rate (66 MHz), while the remaining 1-x are executed at the normal rate (33 MHz). It's observed that the 486DX2-66 is 76% faster than the 486DX-33 (which executes all instructions at 33 MHz). Given this and making some reasonable assumptions, estimate how much faster a clock-tripled 486DX3-99 (on chip 99 MHz; off chip 33 MHz) is than the 486DX2-66.
At Basil's Burgers you can purchase Cheap Chicken Chunks in boxes holding 6, 9 or 20 Chunks. Therefore, you could purchase exactly 21 Chunks by purchasing two boxes of size 6 and one of size 9. Observe that there is no way to purchase exactly 17 Chunks; what is the largest number such that it is impossible to purchase that many Chunks exactly?
Given that n! = 10888869450418352160768000000, find n (remember that n! = 1*2*3* ... *n). There are several ways of doing this - the trick is to find the easiest.