# Previous Puzzles

June, 29th, 2015               Getting 8 people across the river
We have 8 travellers: 1 criminal, 1 police man, 1 mother with her 2 daughters, 1 father with his 2 sons  (2 different families). The mother wants to kill the 2 sons, the father wants to kill the 2 daughters. The criminal wants to kill everybody. The criminal can’t kill anybody if the police man is present. The mother can’t kill the sons if the father is present, the father can’t kill the daughters if the mother is present.  The police man will not prevent the mother or the father from killing any of the children.
All eight travellers arrive at a river at the same time. They want to get across. However, there is only one boat, which can take maximal two people. The boat can only be navigated by an adult.
How can you get all eight people across the river, without anybody getting killed?

July 28th, 2014                                         Petrol

A petrol station has two empty jugs: a three litre jug and a five litre jug. How can he measure exactly one litre without wasting any petrol?

July 21nd, 2014                                         Bridge Crossing

Four people come to a river in the night. There is a narrow bridge, but it can only hold two people at a time. Because it’s night, the torch has to be used when crossing the bridge.
Anyango can cross the bridge in 1 minute, Ben in 2 minutes, Caro in 7 minutes, and Dan in 10 minutes. When two people cross the bridge together, they must move at the slower person’s pace.
The torch must be walked back along the bridge, it cannot be thrown back over.
How quickly can they all get across the bridge?

July 2nd, 2014                                         Buses

A bus route has a total duration of 40 minutes.
Every 10 minutes, two buses set out, one from each end.
How many buses will one bus meet on its way from one end to the other end?

May 29th, 2014                                         Monty Hall

Use 3 playing cards, a Queen and two 5s. The game is won if the queen is chosen.
One student only knows which card is where. Another student points to a card but does not yet turn it over. The student who knows where the cards are, reveals a 5 from the two cards that have not been chosen. Now one 5 is face up and two cards are face down. The student who picked the card is now given the opportunity to change their mind to the other face-down card. If the one they pick is a Queen they win, if it is the 5, they lose.
Repeat this many times and record your results. Is it better to change your mind or not?
You could use columns:
Stay and win              Stay and lose              Switch and win                 Switch and lose

May 22nd, 2014                                         Australian Shuffle Instructions

• Deal 4 piles of 4 cards in a row and then pick up and discretely check the bottom card of one of the pile.
• Mention that you are psychic and can predict things in the future. Now write the card down on a piece of paper and place it face down on the table. Make sure the contestant cannot see what you have written
• Put the 4 piles together with the pile you’ve looked at on the bottom. Then put these cards back on the top of the deck (the card you’ve looked at will be in the 16th position).
• Ask the contestant to cut the cards roughly in two (they need to take between 17 and 31 cards for the trick to work). Ask them to cut the again if it seems wrong (just say “That was never half”)
• Use the top half of the deck and leave the remainder on the table.
Do the Australian shuffle.
• You have to put the 1st card on the remainder pile and the 2nd face up on the table.
• Then the 3rd card on the remainder pile and the 4th face up on the table. Keep doing this until all cards on the table.
Repeat Australian Shuffle
• Pick the face up cards, turn upside down and deal cards again, the first on the remainder pile face down and the 2nd on the table face up.
• Keep repeating (ask the contestant to do it for you) with the face up card pile until there is one card left on the table. (This is the magic card which you have written down).
• Turn the piece of paper and show them the card!
Make a poster showing mathematically why the trick works

May 14th, 2014                                         10 Card trick

•    Ask the volunteer to shuffle the pack
•    Ask the volunteer to choose any 10 cards from the pack face down and put into a pile.
•    Ask them to pick one magic card from the ten cards (remember it) and place on top of the pack of 10 cards. The magician shouldn’t see the card. When practising the trick however, the magician can have a look just to check they are doing the trick correctly.
•    Magician then turns their back and asks the volunteer to choose a secret magic number between 1 and 10.
•    Ask the volunteer to move that many cards, one by one to the back of the 10 card pile (from the top of the pile to the bottom of the pile).
•    Magician the turns round and takes the 10 card pile and looks through the cards face down. Move once card at a time to the other hand. In doing this, you are reversing the order of the cards.
•    The magician then takes the top five cards of the new arrangement (face up) and moves them to the bottom of the pile. Whilst doing this, magician acts like they are trying to mentally find the card. Don’t make it obvious that you are moving the cards around.
•    The magician says the magic just isn’t working and needs the volunteer to do more.
•    Ask the volunteer to take the pile again face down and move the same number of cards as he/she did before, but with one extra card. Magician turns their back and doesn’t watch.
•    Magician then turns round and takes the cards and looking at them face down, picks the fourth card and shows them. This was the magic card!
Make a poster showing mathematically why the trick works

April 22nd, 2014                                         Pair Sums

Five numbers are added together in pairs to produce the following answers:

0,2,4,4,6,8,9,11,13,15

What are the five numbers?

April 15th, 2014                                         14 Divisors
The list below shows the first ten numbers together with their divisors (factors):

1. 1
2. 1, 2
3. 1, 3
4. 1, 2, 4
5. 1, 5
6. 1, 2, 3, 6
7. 1, 7
8. 1, 2, 4, 8
9. 1, 3, 9
10. 1, 2, 5, 10

What is the smallest number with exactly twelve divisors?
What is the smallest number with exactly fourteen divisors?

March 31st, 2014                          Mod 3
Prove that if a2 + b2 is a multiple of 3 then both a and b are multiples of 3.

March 17th, 2014              Double Digit

Choose two digits and arrange them to make two double-digit numbers, for example: If you choose 1 and 2, you can make 12 and 21
Try lots of examples. What happens? Can you explain it?
What happens if you choose zero as one of the digits? Try to explain why.
How does it work if you choose the same digits, for example 3 and 3?
What happens if you use negative numbers?
Now choose three digits and arrange them to make six different triple-digit numbers.
Do you get the same results?
If you’re feeling very organised, try more digits and see what happens.

March 10th,2014              An Appearing Act

Take an 8 by 8 square, and cut it up according to the diagram below.
You may wish to print out some dotty paper (Use grids on a squared paper).
Rearrange the pieces to make a 13 by 5 rectangle:
Why are the two areas not the same?
Can you divide other squares up to perform a similar trick? Explore!
This resource is part of “Dotty Grids – Exploring Area“. Click here to explore.

March 3rd,2014              Achi

A game for two players
You need: Four counters each and A game board
To start:
Take it in turns to place a counter on  an empty circle. Keep going until all the counters are on the board. Then take it in turns to slide one of your counters along a line to an empty circle.
The winner: the first player to get three counters in a straight line. View previous puzzles

February  24th,2014              Square Areas

The outer square has sides of length 1.
Can you work out the area of the inner square and give an explanation of how you did it?

February 17th,2014              Why 8?

Choose any four consecutive even numbers. (For example: 6, 8, 10, 12). Multiply the two middle numbers together. (e.g. 8 x 10 = 80) Multiply the first and last numbers. (e.g. 6 x 12 = 72) Now subtract your second answer from the first. (e.g. 80 – 72 = 8)
Try it with your own numbers. Why is the answer always 8?

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice.Can you find all the ways to do this?

February 3rd,2014           Square Maze

A square maze has 9 rooms with gaps in the walls between them. Once a person has travelled through a gap in the wall it then closes behind them. How many different ways can someone travel through the maze from X to Y?

A cube (1 metre by 1 metre by 1 metre) has one face on the ground and one face pressed up to a wall. A ladder, 4 metres long, is leaning against the wall and it just touches the cube. How high is the top of the ladder above the ground?

For the solution and more puzzles see:
http://nrich.maths.org/289/solution

September 12th,2013
It takes two gardeners 8 days to mow a lawn.
One is lazy and one is energetic.
The energetic one would only take 12 days to mow it on his own.
How many days would the lazy gardener take to mow the lawn on his own?
For the solution and more puzzles see:
http://www.mathsisfun.com/lawn_problem_solution.html

March 31st, 2014                                          Mod 3
Prove that if a2 + b2 is a multiple of 3 then both a and b are multiples of 3.

August 14th, 2013 Puzzle
LOGO Challenge – the Logic of LOGO
Just four procedures were systematically used to produce the design below.

How would you set about the task of deciding what to do? Can you use fewer procedures? How would you replicate this design?
How systematic were you in your endeavours?
For the solution and more problems see Solutions.

Weekly Puzzle: August 6th, 2013 Puzzle
Children at Large
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben.
Tom is 2 years older than Ben. The combined ages of the two boys is equal to the combined ages of the two girls.
Kate is twice as old as Sally. A year ago Tom was twice as old as Sally was then. How old are the children? For the solution and more problems see Solutions.

Weekly Puzzle: July 30th, 2013 Puzzle
Look at the following row of numbers:  10 15 21 4 5

They are arranged so that each pair of adjacent numbers adds up to a square number: 10+15=25; 15+21=36; 21+4=25; 4+5=9

Can you arrange the numbers 1 to 17 in a row in the same way, so that each adjacent pair adds up to a square number? For the solution and more problems see Solutions.

July 23rd, 2013 Puzzle

Sometime during every hour the minute hand lies directly above the hour hand.

At what time between 4 and 5 o’clock does this happen?

For the solution and more problems see http://nrich.maths.org/1782/solution.

February 27th, 2013 Puzzle
Given that 5^j+ 6^k + 7^l + 11^m = 2006 where  j ,k ,l and m are different non-negative integers,   what is the value of j+k+l+m?
For the solution and more problems see http://nrich.maths.org/5712/solution

January 30th, 2013 Puzzle
What is the area of this quadrilateral, in cm2?

For the solution and more puzzles see http://nrich.maths.org/6739/solution

January 23rd, 2013 Puzzle

 WINDING VINE LENGTH There is a tree 20 feet high, with a circumference of 3 feet. A vine starts at the base of the tree and winds around the tree 7 times before reaching the top. How long is the vine?Hint: There is an easy way to solve this problem which only uses junior high school math! Note 1: Apparently from Chinese texts over 2000 years old. Note 2: Treat the tree as a perfect cylinder.

January 9th, 2013 Puzzle

 TRUE OR FALSE LIST Consider a list of 2000 statements:1) Exactly one statement on this list is false. 2) Exactly two statements on this list are false. 3) Exactly three statements on this list are false. . . . 2000) Exactly 2000 statements on this list are false.Which statements are true and which are false?What happens if you replace “exactly” with “at least”?What happens if you replace “exactly” with “at most”?What happens in all three cases if you replace “false” with “true”?

December 12th, 2012 Puzzle

 TWO CONSECUTIVE HEADS An unbiased coin is tossed n times. What is the probability that no two consecutive heads appear?

November 28th, 2012 Puzzle
A patient has fallen very ill and has been advised to take exactly one pill of medicine X and exactly one pill of medicine Y each day, lest he die from either illness or overdosage. These pills must be taken together. The patient has bottles of X pills and Y pills. He puts one of the X pills in his hand. Then while tilting the bottle of Y pills, two Y pills accidentally fall out. Now there are three pills in his hand. Because both types of pill look identical, he cannot tell which two pills are type Y and which is type X. Since the pills are extremely expensive, the patient does not wish to throw away the ones in his hand. How can he save the pills in his hand and still maintain a proper daily dosage?

November 21st, 2012 Puzzle

I took the graph y=4x+7 and performed the four transformations below.
Reflect in the vertical axis
Translate down by three units
Translate left by two units
Reflect in the horizontal axis
Unfortunately, I can’t remember the order in which I carried out the four transformations, but I know that I ended up with the graph of y = 4×-2.
Find an order in which I could have carried out the transformations.
There is more than one way of doing this – can you find them all?
Try to explain why different orders can lead to the same outcome.
What other lines could I have ended up with if I had performed the four transformations in a different order?
For the solution and more puzzles see http://nrich.maths.org/6544/solution

November 7th, 2012 Puzzle
Circle-in
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17.
What is the radius of the circle?
Can you adapt your method to find the radius of a circle inscribed in any right-angled triangle ABC?
For the solution and more puzzles see http://nrich.maths.org/2163/solution

October 28st, 2011 Puzzle

Roll a standard pair of six-sided dice, and note the sum. There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12. Find all other pairs of six-sided dice such that:

1. The set of dots on each die is not the standard {1,2,3,4,5,6}.
2. Each face has at least one dot.
3. The number of ways of obtaining each sum is the same as for the standard dice.

For a hint, solution and many more puzzles see Nick’s Mathematical Puzzles.

October 24th, 2012

 21 FACTORIAL M 4/7/2003 12:17AM 21!=510909x21y1709440000 Without calculating 21!, what are the digits marked x and y?

October 21st 2011 Puzzle

This picture shows the start of Pascal’s triangle.  There are many patterns to be found within Pascal’s triangle.
What does the fourth diagonal show?
Try coloring in multiples of 2, 3, 4, 5 etc and see what patterns emerge.  See this webpagefor an interactive demonstration of this.
Can you describe a pattern in the digits of the horizontal rows?
Can adding up the digits in each row lead you to be able to write down the sum of the digits in the 10th row without working out each digit?
There is a wealth of discussion of Pascal’s triangle on the internet.  After exploring any patterns you can see, you might like to start with mathisfun.

October 17th 2011 Puzzle

On your travels you come to an old man on the side of the road holding three cards from a standard deck face down. Trying to make conversation you ask him what the three cards are.
He tells you, “To the left of the queen, are one or two jacks. To the right of the jack, are one or two jacks. To the right of the club, are one or two diamonds. To the left of the diamond, are one or two diamonds.” What are the three cards?
For the answer and more puzzles see mathisfun webpage.

October 17th, 2012

Funny Factorisation
Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once.
The number can be written as just such a product. Can you find the factors?
Maths is full of surprises! The number can be written as a product like this in two DIFFERENT ways, and so can the number. Can you find these four funny factorisations?
For the solution and more puzzles see http://nrich.maths.org/740/solution

October 7th 2011 puzzle

Imagine three dice that are used to play a game for two players:
The red die has the numbers {1, 1, 6, 6, 8, 8}
The green die has the numbers {2, 2, 4, 4, 9, 9}
The blue die has the numbers {3, 3, 5, 5, 7, 7}
Each player chooses a different die. They roll their dice. The winner is the person whose die shows the bigger number.
Alison and Charlie are playing the game. Charlie wants to go first so Alison lets him.
Was that such a good idea?
Can you advise Alison on which die to choose once she knows which die Charlie has selected?
These types of dice are called transitive.  Go to nrich for the solution and for more information on transitivity.

October 10th, 2012

How many steps are required to break an m x n bar of chocolate into 1 x 1 pieces?
You can break an existing piece of chocolate horizontally or vertically.
You cannot break two or more pieces at once (so no cutting through stacks).
For solution see http://www.mathsisfun.com/puzzles/breaking-up-a-chocolate-bar.html

September 26th, 2012

 SUM OF REAL NUMBERS 5/11/2003 10:59PMThe sum of N real numbers (not necessarily unique) is 20. The sum of the 3 smallest of these numbers is 5. The sum of the 3 largest is 7. What is N?