Scales and Vessels
How can you measure out
exactly 4 litres of water from a tap using a 3 litre and a 5 litre bucket?
A 24 litre bucket is
full of lemonade. 3 men 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?
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?
One of 9 otherwise
identical balls is overweight. How can it be identified after 2 weighings with
an old balance?
One of 27 otherwise
identical balls is overweight. How can it be identified after 3 weighings with
an old balance?
How many ways can you
put 10 sweets into 3 bags so that each bag contains an odd number of sweets?
Ferries
A man has to take a
hen, a fox, and some corn across a river. He can only take one thing across at
a time. Unless the man is present the fox will eat the hen and the hen eat the
corn. How is it done?
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?
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.
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?
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?
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".
Each letter represents
a different digit
SEND
+MORE
-----
MONEY
This sum uses all the
digits
28*
+**4
----
****
This subtraction sum
uses all the digits from 1 to 9.
9 * *
- * 4 *
-----
* * 1
O represents odd digits
E represents even digits
EEO
xOO
-----
EOEO
EOO
-----
OOOOO
P represents prime
digits
PPP
xPP
-----
PPPP
PPPP
-----
PPPPP
Some more additions
THE
TEN
MEN
----
MEET
SLOW
SLOW
OLD
----
OWLS
SAL
SEE
THE
SUEZ
-----
CANAL
FIVE
FIVE
NINE
ELEVEN
------
THIRTY
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
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?
The product of 3
brothers' ages is 175. Two are twins. How old is the other one?
A man 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?
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?
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?
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)
Make fractions (each
using all the digits from 1 to 9) with these values 1/2, 1/3....1/9
A greengrocer was
selling apples at a penny each, bananas at 2 for a penny and pears at 3 for a
penny. A father spent 7p and got the same amount of each type of fruit for each
of his 3 children. What did each child get?
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?
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?
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?
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?
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?
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?
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.
How many presents did
the "true love" send during the 12 days of Christmas?
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?
Enumerations
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?
You have 3 bricks, each
measuring 18 x 9 x 6 cm. How many different heights can you build up with them?
How many right-angled
triangles with integral sides have one side of 15?
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?
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?
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?
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
See SMP E for answers and developments
Geometry
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.
How many straight lines
of unique lengths can you draw from dot to dot?
How many non congruent
triangles can you draw?
Miscellaneous
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?
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?
The bacteria in a
test-tube double each minute. It is full in one hour. When was it half full?
In a knockout with 39
players, how many games are played?
In a knockout with 39
players, how many byes?
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?
If a crocus costs 8p, a
tulip 7p and a daffodil 11p, how much does a hyacinth cost?
Hint: count the vowels and consonants
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.
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).
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
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?
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.
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
A car milometer shows
15951. After 2 hours it shows another palindromic number. How fast was the car
going on average?
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).
NIM
Players
take turns to remove any number of objects from one of n rows. The player who
takes the last object wins.
Ans:If
a game is played with a finite set of numbers and each move changes only one
set and the game terminates, it is equivalent to NIM.
To determine who
should win, write the sizes of the piles in binary. Add the columns (no
carrying). If the sum of each column is even, the person to play loses. (from
Mathematical Carnival)
In
the reverse game the player who takes the last counter loses
Ans: The strategy in the reverse game follows the normal game until only one row has more than one counter. Then you take away either all the biggest row or all but one of this row so as to leave an odd number of rows.
Variations include
PRIM - Remove from a
heap of n any prime
DIM - Remove a divisor
of n from a heap of n
Remove any square
number from a heap.
RIMS - Draw some dots.
Players take turns to join them up. No loops are allowed to overlap. Dots
within topological regions give a NIM game.
Sprouts - Draw a few
(3 is enough) dots on a piece of paper. Players take turns to join one dot to
another or itself, then put a new dot on the line. A line can't cross itself or
another line, nor can it pass through a dot. No spot can have more than 3 lines
going to it. The winner is the person who makes the last move. (from
Mathematical Carnival)
Rayles - same as RIMS
except that only 1 or 2 spots can be joined.
Kayles (isomorphic to
Rayles) Skittles are lined up. Can knock down two adjacent skittles, or one
skittle. The last player wins.
Grundy - Split any
heap into 2 non-empty heaps.
Noughts and Crosses
In a variation of the
game, where the first person to make a line loses, the first player can only
force a draw if they first play in the centre (in "Mathematical puzzles
& diversions")
Make 9 cards with the
digits 1-9 on them (one per card). Scatter the cards face-up. Players take turns
picking a card. The 1st to have 3 cards which add up to exactly 15 wins.
Ans: this game is isomorphic to Noughts and Crosses where the cells have the following values
2 9 4
7 5 3
6 1 8
Make 9 cards with the
following words on them (one per card) HOT FORM WOES TANK HEAR
WASP TIED BRIM SHIP. Scatter the cards face-up. Players
take turns picking a card.
The 1st to hold 3 cards with same letter on wins.
Ans: this game is isomorphic to Noughts and Crosses where the cells have the following values
HOT FORM WOES
TANK HEAR WASP
TIED BRIM SHIP
Grid Games
Draw a 3x3 grid.
Players can color in any amount of squares on one row or column. The winner is
the one who fills in the last square.
Ans: The 2nd player can always win. If the first player fills a single cell in, then the 2nd player should fill 2 cells in to make an 'L'. Otherwise the 2nd player should complete an 'L' or 'T' consisting of 5 cells.
Ans: The 2nd player can always win. If the first player fills a single cell in, then the 2nd player should fill 2 cells in to make an 'L'. Otherwise the 2nd player should complete an 'L' or 'T' consisting of 5 cells.
TacTix - Draw a 4x4
grid. Players can color in any amount of adjoining squares on one row or
column. The winner is the one who fills in the last square. Ans:
Gardner in "Mathematical puzzles & diversions" says that the
second player can always win, but there's no simple strategy. The reverse game
is much harder. A variation, where the colored in squares needn't be adjoining,
is harder still.
Miscellaneous
10x10
2 Dice. Shade in a
rectangle of size X by Y,
where X and Y are
die scores. If X=Y, the player can shade
in X*Y squares
anywhere (and needn't finish exactly)
1 die. Fill in a
rectangle with the area.
2 players take turns
to add 1-10 to a common total. First to 100 wins.
Ans: (First to 89 wins)
Tug of war
| | | |X| | | |
Each player has 50
muscle points to gamble. At each turn the one who gambles the most pulls the
knot one square their way. The player who pulls the knot to their end wins.
13 petals (Sam Lloyd)
Remove 1 or 2 connected petals. The one who takes the last wins.
Ans: (preserve symmetry)
References
On Games and Numbers,
J.H. Conway
Mathematical Carnival,
Martin Gardner, Penguin 1976
Mathematical puzzles
& diversions, M. Gardner, London, 1961.
Mathematical Games, C.
Lukacs and E. Tarjan, Souvenir, 1969
References
Mathematical puzzles and pastimes, A. Bakst, Princeton,N.J, 1965.
More mathematical activities: a resource book for teachers, Brian Bolt, Cambridge University Press, 1985.
The Corgi Book of Problems, Jameson Erroll, Transworld Publishers Ltd., 1964.
Mathematical Bafflers, Angela Dunn, Dover Publications, 1980.
Mathematical puzzles & diversions, M. Gardner, London, 1961.
More mathematical puzzles & diversions, M. Gardner, London, 1963.
Mathematical puzzles for the connoisseur, P.M.H. Kendall and G.M. Thomas, London, 1962.
Mathematical games and puzzles, T. Rice, London, 1973.
The Canterbury Tales, Henry Dudeney, Thomas Nelson and sons, 1949
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