Problems for Metagrobologists

A Collection of Puzzles with Real Mathematical, Logical or Scientific Content

David Singmaster

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789814663663
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

This book is a collection of over 200 problems that David Singmaster has composed since 1987. Some of the math problems have appeared in his various puzzle columns for BBC Radio and TV, Canadian Broadcasting, Focus (the UK popular science magazine), Games and Puzzles, the Los Angeles Times, Micromath, the Puzzle a Day memo pad and the Weekend Telegraph. While some of these are already classics, many of the puzzles have not been published elsewhere previously.

Puzzle enthusiasts of all ages will find here arithmetic problems, properties of digits; monetary problems; alpha-metics; Diophantine problems; magic figures; sequence problems; logical problems; geometric problems; physics problems; combinatorial problems; geographic problems; calendar problems; clock problems; dissection problems and verbal problems.

Contents:

• General Arithmetic Puzzles
• Properties of Digits
• Magic Figures
• Monetary Problems
• Diophantine Recreations
• Alphametics
• Sequence Puzzles
• Logic Puzzles
• Geometrical Puzzles
• Geographic Problems
• Calendrical Problems
• Clock Problems
• Physical Problems
• Combinatorial Problems
• Some Verbal Puzzles

Readership: General public.
Metagrobologists;Alphametics;Magic Figures;Clock Problems;Diophantine Key Features:

• The problems are generally original, though some are corrections or extensions of known problems
• A number are open-ended, leading to unsolved problems for the reader

Chapter 1

General Arithmetic Puzzles

1.SHARE AND SHARE ALIKE

Jessica and her friend Pud like to eat a big lunch. One day Jessica brought four sandwiches and Pud brought five. Samantha got mugged on her way to school, but the mug ran off with her lunch and left her purse. So Jessica and Pud shared their sandwiches with Samantha. After eating, Samantha said: “Thanks a million. I’ve got to see Mr. Grind, but here’s some money to pay for the sandwiches.” She left $3 and ran off. Jessica said: “Let me see, I brought four and you brought five, so I get 4/9 of $3, which is 4/3 of a dollar, which is $1.33, near enough.” Pud said: “Ummm, I’m not sure that’s fair.” Why?

Based on Fibonacci (1202)

2.LEO’S LILLIAN LIMERICK

Leo Moser was a prolific mathematician who vastly enjoyed the subject. His brother Willy kindly sent me a fine example of Leo’s mathematical limericks. I (with much help from my wife) have extended this to clarify the problems.

Once a bright young lady called Lillian
Summed the NUMBERS from one to a billion
But it gave her the “fidgets”
To add up the DIGITS.
If you can help her, she’ll thank you a million.
If you are as bright as this Lillian,
Sum the NUMBERS from one to a billion
And to show you’re a whiz
At this kind of biz,
Sum the DIGITS in numbers one to a billion.

3.SOME SQUARES

My friends, Mr. and Mrs. Able and Mr. and Mrs. Baker, went shopping for Christmas presents. They split up and later rejoined for lunch, each having bought something. Mrs. Able said: “An odd thing happened this morning. I bought as many items as each item cost in dollars.” Amazingly, each of the others said they had done the same thing, though they had all bought different numbers of items. Even more amazingly, the Ables had spent just as much as the Bakers. What is the smallest amount they could have spent?

4.SOME PRODUCT

My secretary, Ms. Flubbit, has gone to visit her relatives and Mr. Flubbit is standing in for her. He has been upholding the family tradition. I asked him to add up some positive whole numbers and he managed to multiply them instead! But when I did the sum, I got the same result as he had! I told my daughter Jessica about this remarkable occurrence. After some thought, she said: “I bet I can tell you the numbers if you tell me how many numbers there were.” But I was way ahead of her and responded: “It’s a good bet to try, but you’ll lose!” What is the least number of numbers there could have been and how many sets of numbers could there be in this case?

5.SUM TROUBLE

Jessica and her friend Hannah were looking at a puzzle book which asked how to put + and − signs into the sequence 123456789 in order to make it add up to 100. After a bit, they looked at the answer and found 1 + 2 + 3 − 4 + 5 + 6 + 78 + 9 = 100. Hannah said: “I bet there are more ways to do this.” Jessica said: “Sure, but I don’t like that minus sign, it’s too complicated. I’d rather have only plus signs. I wonder if that’s possible.” “If it were possible, they’d have asked for it”, replied Hannah. “Possibly. Or perhaps it’s too easy”, mused Jessica.

Who is right?

6.RUSH AND SEDGE

Here is a problem from the early Chinese classic Chiu Chang Suan Ching (Nine Chapters on the Mathematical Art), which is variously dated between −2nd century (i.e. the second century BC) and +2nd century. Most of its problems are familiar and many traveled via India and the Arab world to medieval Europe. The following is an example which does not seem to have been copied elsewhere.

A rush grows 3 feet on the first day, then 3/2 feet on the second day, 3/4 foot on the third day, . . . . A sedge grows 1 foot on the first day, 2 feet on the second day, 4 feet on the third day, . . . . When are they the same size?

The Chiu Chang Suan Ching gives the answer 2 6/13 days. Can you see how this was obtained? Can you find a more correct solution?

7.SHE’S A SQUARE

Jessica’s friend Katie says she will be x years old in the year x². How old will she be this year?

8.LEMONADE AND WATER

Jessica had a jug containing 7 pints of lemonade mixed with 11 pints of water. Her friend Rachel had a jug with 13 pints of lemonade mixed with 5 pints of water. Jessica poured two pints from her jug into Rachel’s jug. After mixing it thoroughly, Rachel poured two pints back into Jessica’s jug. They repeated this three more times. How much more lemonade has Jessica gained than Rachel has gained water?

9.AN OLD MISTAKE

A problem book from the turn of the century asks: “What is the first term of an arithmetic progression of five terms if the sum of all five terms is 40 and their product is 12320?” The book gives the answer 3. Do you agree? What question could this be the answer to?

10.HORSE TRADING

I took Jessica and her friend Hannah to a country horse fair, where we watched a trader in action. In the morning he bought a batch of similar horses. Looking them over, he decided that one was a bit better than the others and he would keep it. In the afternoon he sold the others at £20 more per horse than he had paid per horse. After some figuring, Jessica and Hannah realized that he had received just as much as he had paid out for all the horses in the morning. Jessica said: “So he would also have gotten as much money if he kept two horses and sold the others for £40 per horse more than he had paid.” Hannah said: “No, I figure he would have to get £45 more per horse.” I said: “One of you is right.”

Who is right? And how many horses did the trader buy and how much did he pay for each horse?

11.A WEIGHTY PROBLEM

An old book of games and puzzles gives the following problem and answer.

“A man had a set of weigh-scales and only four weights, yet with those four weights he could weigh any number of pounds up to [and including] forty. What were the four weights?

The four weights were, 3 lb, 4 lb, 6 lb, and 27 lb.”

Obviously our man works in a cheese factory where he wants to know how many ways he can weigh his whey. What weights of whey can he weigh? I thought that perhaps the 6 lb was a misprint for 9 lb. If he has 3, 4, 9 and 27 lb weights, is he better off and in what way?

[This problem is often known as Bachet’s weight problem since Bachet described it in 1612, but it was described already by Fibonacci in 1202. Fibonacci gives a set of four weights that allows you to weigh any number of pounds up through forty pounds. You probably know the standard answer. If not, try to find such a set of weights. If you do know the answer, can you prove that it is unique?]

12.TRIANGULATION

Jessica and Rachel have been marble fanatics for the last few months. They have been playing as a team and have managed to win almost all the marbles in the neighborhood. Jessica was laying out the marbles on the floor and she started putting them into a triangular form, like the 10 pins in bowling or the 15 balls in pool or snooker. They were surprised to find that the marbles made a perfect triangle. Rachel then arranged them into smaller triangles and they were even more surprised to find that the marbles made an exact number of equal triangles. They then tried other sizes of triangles and they found eight ways altogether to form the marbles into an exact number of equal triangles. (Of course, this counted the first case when there was just one triangle.) They didn’t have a huge number of marbles — certainly less than a thousand or two — so how many did they have?

13.MATCHSTICKS OR FIDDLESTICKS

By mixing Roman numerals, 1, and the arithmetic operators +, −, × and /, one can form various numbers with five matchsticks. For example: IV/I = 4, V × I = 5. Make all the integer values from 0 through 17. I haven’t been able to make 18 — can you?

14.CALCULATED CONFUSION

Jessica likes playing with her calculator and our computer. The other day she discovered that they don’t behave the same way. She put in 3 + 4 × 5 into her calculator and got 35, but when she put it on the computer, it gave 23. The calculator (if it is old enough) does each operation immediately, so it added 3 to 4 to get 7 and then multiplied this result by 5 to get 35. The computer looks at the whole expression and evaluates products before sums, so it multiplied 4 by 5 to get 20 and then added 3 to get 23. After I explained this to Jessica, she went away and did a lot of button punching. She came back after a bit and said she had been looking for an example where the two methods of calculation gave the same result, but she always found the first method gave a larger answer than the second method. Does this always happen? If not, when can the first result be the smaller? Can the two results ever be equal? If so, when? Why didn’t Jessica find such examples?

15.JELLY BEAN DIVISION

Jessica and her neighbors Hannah and Rachel bought a big bag of jelly beans. Because they had put in different amounts of money, they wanted to count the number of jelly beans in order to share them, but they kept losing count. All they could establish was that there were less than 500. Finally Jessica decided to count them out by 7s. She found that there were just three jelly beans left over. By now they were hungry, so they agreed to eat the three. Hannah then counted out the beans by 8s. She found there were three jelly beans left over and they ate them. Rachel then counted them out by 9s and, to her surprise, she found there were three jelly beans left over and they ate them. How many beans were there at the beginning?

16.SONS AND DAUGHTERS

Reading a modern puzzle book from India, I found a somewhat cryptic version of a standard problem.

A man died leaving an estate of 1,920,000 rupees with the proviso that each son should receive three times as much as a daughter and each daughter should receive twice as much as a mother. How much did the mother receive?

Clearly some information has been lost, so I looked at the answer, which was 49,200 10/13 rupees. A little work shows that this cannot be correct. Why? What should the answer be? And how many sons and daughters might there be?

17.THREE BRICKLAYERS

I wanted a wall built in my backyard. I asked my local builder for an estimate and especially wanted to know how long it would take. He said he’d previously built several walls in the road that were just the same size. But he had three bricklayers and he’d only ever used them in pairs. When he’d put Al and Bill on the job, they did it in 12 days, but Al and Charlie took 15 days while Bill and Charlie took 20 days. So I asked how long it would take each man working alone. That stumped him but he thought he could work out how long it would take all three together but he didn’t have his calculator with him. Can you find all these times for us? If you find this classic problem too easy, can you explain how to find numbers so that all the times turn out to be whole numbers?

18.AN ODD AGE PROBLEM

Jessica is just 16 and very conscious of her new age. Her neighbor Helen is just 8 and I was teasing Jessica. “Seven years ago, you were 9 times as old as Helen; six years ago, you were 5 times her age; four years ago, you were 3 times her age; and now you are only twice her age. If you are not careful, soon you’ll be the same age!” Jessica seemed a bit worried and went off muttering. I saw her doing a lot of scribbling. Next day, she retorted, “Dad, that’s just the limit! By the way, did you ever consider when I would be half as old as Helen?” Now it was my turn to be very worried and I began muttering — “That can’t be, you’re always older than Helen.” “Don’t be so positive,” said Jessica as she went off to school. Can you help?

19.SOME SQUARE SUMS

You undoubtedly know the idea of a magic square. It’s an n by n array of the first n² integers such that each row, each column and the two diagonals total up to the same value, which is n(n² + 1)/2. There is essentially just one magi...

Table of contents

• Cover page
• Title page
• Copyright page
• Dedication
• Contents
• Introduction
• Chapter 1 General Arithmetic Puzzles
• Chapter 2 Properties of Digits
• Chapter 3 Magic Figures
• Chapter 4 Monetary Problems
• Chapter 5 Diophantine Recreations
• Chapter 6 Alphametics
• Chapter 7 Sequence Puzzles
• Chapter 8 Logic Puzzles
• Chapter 9 Geometrical Puzzles
• Chapter 10 Geographic Problems
• Chapter 11 Calendrical Problems
• Chapter 12 Clock Problems
• Chapter 13 Physical Problems
• Chapter 14 Combinatorial Problems
• Chapter 15 Some Verbal Puzzles
• Solutions