# Mathematical Puzzles

These puzzles (most of them old classics) from various sources can be used with pupils who finish classwork early. Most of the questions were chosen with enthusiastic, bright early teenagers in mind.

Some of the puzzles are also appropriate for class work - an initial worked example on the board will help a lot.

There are a few trick questions. Some questions can be quickly answered if you chance upon the right approach, but the 'long' solution isn't too arduous. Several of the questions are best answered by writing a computer program. Some are shown here.

## Scales and Vessels

1. How can you measure out exactly 4 litres of water from a tap using a 3 litre and a 5 litre bucket?
2. A 24 litre bucket is full of lemonade. 3 people want to have equal amounts of it to take home, but they only have a 13 litre, a 5 litre and an 11 litre bucket. How do they do it?
3. A Queen (78kg), the Prince (36kg) and the King (42kg) are stuck at the top of a tower. A pulley is fixed to the top of the tower. Over the pulley is a rope with a basket on each end. One basket has a 30kg stone in it. The baskets are enough for 2 people or 1 person and the stone. For safety's sake there can't be more than a 6kg difference between the weights of the baskets if someone's inside. How do the people all escape?
4. One of 9 otherwise identical balls is overweight. How can it be identified after 2 weighings with an old balance?
5. One of 27 otherwise identical balls is overweight. How can it be identified after 3 weighings with an old balance?
6. How many ways can you put 10 sweets into 3 bags so that each bag contains an odd number of sweets?

## Ferries

1. A farmer has to take a hen, a fox, and some corn across a river. The farmer can only take one thing across at a time. Unless the farmer's present the fox will eat the hen and the hen eat the corn. How is it done?
2. 3 missionaries and 3 obediant but hungry cannibals have to cross a river using a 2-man rowing boat. If on either bank cannibals outnumber missionaries the missionaries will be eaten. How can everyone cross safely?
3. 2 men and 2 boys need to cross a river in a boat big enough for 1 man or 2 boys. How do they do it?
SMP and CSE 1974 extend this to cover the case of n men.

## Picking Captains

1. 6 girls pick a captain by forming a circle then eliminating every n'th girl. The 2nd girl in the counting order can choose n. If she wants to be captain what's the smallest n she should pick?
2. 12 black mice and 1 white mouse are in a ring. Where should a cat start so that if he eats every 13th mouse the white mouse will be last?
3. 20 passengers are in a sinking ship. 10 are mathematicians. They all stand in a ring. Every 7th climbs into the lifeboat which can only hold 10 people. Where should the mathematicians stand in the ring?
4. 30 passengers are in a sinking ship. They all stand in a circle. Every 9th passenger goes overboard. The lifeboat holds 15. Where are the 15 lucky positions in the circle?

## Incomplete Sums

Some worked examples are in J.A.H. Hunter's "Mathematical Brain Teasers".

1. Each letter represents a different digit
```      SEND
+MORE
-----
MONEY
```
2. This sum uses all the digits
```    28*
+**4
----
****
```
3. This subtraction sum uses all the digits from 1 to 9.
```  9 * *
- * 4 *
-----
* * 1
```
4. O represents odd digits E represents even digits
```  EEO
xOO
-----
EOEO
EOO
-----
OOOOO
```
5. P represents prime digits
```  PPP
xPP
-----
PPPP
PPPP
-----
PPPPP
```
6. If each letter in the calculation below represents a unique digit from 0 to 9, how many solutions are there?
```
SIX
x TWO
-----
DOZEN
```
7. Some more additions
• ``` THE
TEN
MEN
----
MEET
```
• ```SLOW
SLOW
OLD
----
OWLS
```
• ```  SAL
SEE
THE
SUEZ
-----
CANAL
```
• ```  FIVE
FIVE
NINE
ELEVEN
------
THIRTY
```
8. What 5 digit number (where the digits are all different and none of them is zero) multiplied by 4 gives an answer where the digits are those of the original number but in reverse order?

## Letters

Agree on a font of capital letters.
• Put the letters into sets according to line symmetry
• Put the letters into sets according to rotational symmetry
• Put the letters into sets according to topology
• How many letters only use straight lines?
• There is only one number whose English name uses as many straight lines to write as the number itself.
• Think of a number. Write it out in words. Write in words the number of letters you've used (E.g. SIXTEEN-SEVEN-FIVE-FOUR). Continue do so and see what happens. Try 3 other numbers.

## Numbers

1. Alan, Bill and Chris dug up 9 nuggets. Their weights were 154, 16, 19, 101, 10, 17, 13, 46 and 22 kgs. They took 3 each. Alan's weighed twice as much as Bill's. How heavy were Chris's nuggets?
2. The product of 3 sisters' ages is 175. Two are twins. How old is the other one?
3. A person has 2 bankcards, each with a 4 digit number. The 1st number is 4 times the 2nd. The 1st number is the reverse of the 2nd. What is the first number?
4. Tom has 7 sandwiches, Jan has 5, Simon has none. They share them out equally. Simon leaves, paying for his sandwiches by leaving 12 biscuits. What's the fairest way for Tom and Jan to share out the biscuits?
5. A cyclist buys a cycle for 15 pounds paying with a 25 pound cheque. The seller changes the cheque next door and gives the cyclist 10 pounds change. The cheque bounces so the seller paid his neighbour back. The cycle cost the seller 11 pounds. How much did the seller lose?
6. Using four "4"s and common symbols (including the square root, factorial and recurring decimal symbols), make sums whose answers are 0, 1, 2....100 (See Mathematical Bafflers)
7. Make fractions (each using all the digits from 1 to 9) with these values 1/2, 1/3....1/9
8. A greengrocer was selling apples at a penny each, bananas at 2 for a penny and pears at 3 for a penny. A shopper spent 7p and got the same amount (greater than 0) of each type of fruit for each of their 3 children. What did each child get?
9. A woman bought something costing 34c. She only had 3 coins: \$1, 2c and 3c. The shopkeeper had only 2 coins: 25c and 50c. Fortunately another customer had 2 10c coins, a 5c coin, 2 2c coin and a 1c coin. How did they sort things out?
10. Mr and Mrs A are 120 km apart. A bee is on Mr A's nose. The couple cycle towards each other, Mr A at 25km/h and Mrs A and 15km/h. The bee dashes from Mr A's nose to Mrs A's nose and back again and so on at 60km/h. How far does the bee travel before the cyclists crash?
11. Pick a number. If it's even, divide by 2. If it's odd multiply by 3 and add 1. Continue this until you reach "1". Eg 3-10-5-16-8-4-2-1. Which integer less than 100 produces the longest chain?
12. Pick a number. Multiply the digits together. Continue until you get a single digit. What is the only 2 digit number which would require more than 3 multiplication?
13. Starting with 1, place each integer in one of 2 groups so that neither contains a 3 term Arithmetic Progression. How far can you go?
• Apples are packed in boxes of 8 and 15. What is the biggest number of apples that would require loose apples?
• A country only has 5p and 7p coins. Make a list of prices that you could give exact money for. What is the highest prices that you couldn't give exact money for?
14. If D = the day (1-366) in year Y, then the day of the week can be calculated using
```   d = D+Y+(Y-1)/4 - (Y-1)/100 + (Y-1)/400 mod7
```
where d=1 would mean Sunday, etc. Can the first day of each century (e.g. 1st Jan 2001, 1st Jan 1901) be any day?
15. Pick 3 digits (not zero) and make 6 2-digit numbers from them. Add up all these numbers, add up all the original digits and divide the first total by the second.
16. How many presents did the "true love" send during the 12 days of Christmas?
17. At a fairground stall there are 3 piles of cans. You get 3 throws. You can only knock off the top can of a pile. The 2nd throw counts double, the 3rd triple. How do you get exactly 50?

18. If you add the digits in the 1st, 3rd, 5th etc positions of a natural number, and get the same total as you get by summing the other digits, the number is exactly divisible by 11. For example, 17248 is a multiple of 11 because 1+2+8 equals 7+4. But the opposite isn't true - there are multiples of 11 for which the sums don't match. What's the smallest natural number for which this is so?
19. 2 people set out on a journey. One of them walks at a constant 4km/hour. The other walks at 2km/hour for the 1st kilometre, 3km/hour for the 2nd kilometre, and so on. When does the 2nd person catch up with the 1st?

## Enumerations

1. Holding its hands out, palms upward a child starts counting on all its fingers and thumbs, going to and fro. If it counts up to 1982 which finger does counting end on?
2. You have 3 bricks, each measuring 18 x 9 x 6 cm. How many different heights can you build up with them?
3. How many right-angled triangles with integral sides have one side of 15?
4. Minibuses seating 10, 12 and 15 passengers can be used to convey 120 passengers. There are 5 of each size of bus. How many different ways can the buses be used so that all the ones used are full? Which way uses the least buses?
5. Roosters cost 5 pounds, hens 3 pounds and 3 chicks cost 1 pound. Buying at least one of each type of bird how can you buy 100 birds for exactly 100 pounds?
6. A woman puts 120p on the counter. "some 4p stamps, 6 times as many 2p stamps and the rest in 5p stamps please". What does she need?
7. A cube is painted white and cut into 27 small cubes. How many of these little cubes are painted on i) 1, ii) 2, iii) 3, iv) 0 sides?
See SMP E for answers and developments

## Geometry

1. 9 geraniums have to be planted so that there are 3 plants in each row. How can they be planted so that there are a) 8 b) 9 c) 10 lines.

2. • How many straight lines of unique lengths can you draw from dot to dot?
• How many non congruent triangles can you draw?

## Miscellaneous

1. I have some square tiles. Half of them are black and the other ones are white. I arrange them so that the black ones form a rectangle and the white ones make a border around them one tile thick. I have no tiles left over. How many tiles do I have?
2. A man buys 5 cigarettes a day for 25 days. He keeps the stubs. From 5 stubs he made a new cigarette. How many cigarettes does he smoke during this period?
3. The bacteria in a test-tube double each minute. It is full in one hour. When was it half full?
4. In a knockout with 39 players, how many games are played?
5. In a knockout with 39 players, how many byes?
6. A necklace is made from 4 chains each of 3 links. It costs 1p to cut a link and 2p to resolder it. How can a necklace be made for 9p?
7. If a crocus costs 8p, a tulip 7p and a daffodil 11p, how much does a hyacinth cost?
Hint: count the vowels and consonants
8. Arrange a line of 5 coins as follows
``` Head Head Head Tail Tail Tail
```
In 3 moves (each move consisting of turning over a pair of adjacent coins), arrange the coins so that Heads and Tails alternate.
9. Town A and town B are 99 km apart. There are 98 km posts between them. On each post is the distance to A and the distance to B. On how many posts are only 2 different digits used? (for example, the post that says 'A 33, B 66' only uses 2 digits).
10. Clocks -
• How many times is a stopped clock correct each week?
• A clock goes at the right speed, but backwards. How many times each week does it tell the right time?

## Knights

1. If you follow the jumps of a chess knight from word to word you can make a 20 word sentence about the ages of Sue and Sal. Sue is in her teens, so how old is Sal?

2. On a 5 by 5 chessboard, a knight on the centre square would have 8 possible moves. Draw a 5 by 5 grid and put "8" in the middle square. Fill in the other squares with the number of moves that a knight would have from that square.
3. Draw a 5 by 5 chessboard. Put a "0" in the top left corner square. Imagine a knight there. Put a "1" in all the squares that it can reach in 1 go, a "2" in all the squares that it can reach in a minimum of 2 goes, etc, until all the squares are filled in. Which squares take the longest to reach?

## Digital Displays

``` _     _  _       _   _  _   _   _
| | |  _| _| |_| |_  |_   | |_| |_|
|_| | |_  _|   |  _| |_|  | |_|  _|

```
• Make a table showing how many lines are used for each digit.
• While changing from 0 to 1, four bars are turned off. No new ones are turned on. Make a table of what happens as the digits change
```      ON   OFF
0->1  0     4
1->2
...
9->0
```
• Add up the "OFF" and the "ON" columns. What do you notice?

## Palindromes

1. A car milometer shows 15951. After 2 hours it shows another palindromic number. How fast was the car going on average?
2. Pick a 2 digit number. Reverse it then add to the original number. Have you got a palindrome? If not then repeat the process. Try all the 2 digit numbers (as a class activity) (from Mathematical Bafflers).

## References

1. "Mathematical puzzles and pastimes", A. Bakst, Princeton,N.J, 1965.
2. "More mathematical activities: a resource book for teachers", Brian Bolt, Cambridge University Press, 1985.
3. "The Corgi Book of Problems", Jameson Erroll, Transworld Publishers Ltd., 1964.
4. "Mathematical Bafflers", Angela Dunn, Dover Publications, 1980.
5. "Mathematical puzzles & diversions", M. Gardner, London, 1961.
6. "More mathematical puzzles & diversions", M. Gardner, London, 1963.
7. "Mathematical puzzles for the connoisseur", P.M.H. Kendall and G.M. Thomas, London, 1962.
8. "Mathematical games and puzzles", T. Rice, London, 1973.
9. "The Canterbury Tales", Henry Dudeney, Thomas Nelson and sons, 1949
10. "A freshman seminar on problem solving and algorithmic thinking", Edmund A. Lamagna, JCSC 29, 6 (June 2014)
See also Mathematical games
Updated November, 2015
tpl@eng.cam.ac.uk