Problem Solving

Below you will find a selection of my favourite problems gathered from many sources over many years. I hope you or your students enjoy solving these problems. Remember questions are more important than answers, but you may email me if you want any solutions. Enjoy!

#1 Big Sum

What is the sum of all the whole numbers from one to one million?


[Show a simple method to obtain this answer quickly.]

#2 The Alphabet

Pretend that an A is worth one cent, B is worth two cents, C is worth three cents, and so on.

1.  How much is the whole alphabet worth?

2.  What is the value of DECEMBER?

3.  Find a word worth exactly one dollar.

[must be a word found in an English dictionary]


Artwork created by In the Picture Design.

#3 Rectangles

How many rectangles (remember squares are rectangles too!) are there in this diagram?


#4 Dragon Flies

A bat ate 1050 dragon flies on four consecutive nights. Each night she ate 25 more than on the night before. How many did she eat each night? Solve this algebraically.


Image from How Stuff Works.

#5 Shopping

Micky and her Mum are going shopping on Saturday. They bought at least one item from each of the 3 departments that they visited. Micky paid $120 and she got back $11.76 change. What items did they buy?


Dishtowels: $11.38
Curtain Rods: $12.98
Bath Mats: $29.58


Shirt: $30.98
Dress: $49.90
Slacks: $39.90


Hammer: $17.90
Saw: $23.90
Drill: $25.78

#6 The Cube

Look at the picture below of the cube made up of blocks. Each of the blocks in the cube has a different number. Some of the numbers are not visible.

  • Numbers 9 and 13 are directly under number 16.
  • Number 22 is between numbers 9 and 6.
  • Number 17 is next to number 5 and 13, but it is not next to number 19.
  • Number 15 is next to number 24 and number 27.
  • Number 20 is above number 15.

What number block is in the middle of the cube?

Cube Puzzle

#7 Married Couples

Two-thirds of the men in a small country town are married to three-fifths of the women.

How many adults live in this town?

Explain your reasoning.


#8 Ali’s Blocks

StaKubesAli has four 1 centimetre long blocks, three 5 centimetre long blocks, and three 25 centimetre long blocks.

By joining these blocks to make different total lengths, how many different lengths of at least 1 centimetre can Ali make?

Image from highhopes.

#9 Apples for Sale

A fruit stand buys apples at 3 for $1. They will sell them at 5 for $2. How many apples must they sell to make a profit of $100?


#10 Cake Thieves

Mrs Smith baked some beautiful cakes. When she wasn’t looking, some cakes were eaten by one or more of her five children.

When questioned they gave the following answers:

Andrew:  One of us ate the cakes.

Bob:  Two of us ate the cakes.

Joshua:  Three of us ate the cakes.

Lachlan:  Four of us ate the cakes.

Talea:  All of us ate the cakes.

Mrs Smith knew from past experience, that the guilty ones always lied, while the others told the truth.

What was the number of children who ate the cakes?


#11 Fence Posts

Nathan sets fence posts for a living. He takes a job to enclose a rectangular paddock 20 metres by 60 metres. Posts are set 4 metres apart. He charges $12 for setting in each post. What does Nathan charge for this job?


#12 Fruit Picking

Melinda reaches three times into a fruit basket containing 4 apples, 4 bananas, and 4 oranges to get three pieces of fruit.

(a)  How many different collections (combinations) of fruit could Melinda get?

(b)  What is the probability that Melinda will get 3 of the same fruit?


#13 Ship at Anchor

A ship is at anchor. Over its side hangs a rope ladder with rungs 30cm apart. The tide rises at a rate of 28 cm per hour. At the end of 6 hours how much of the rope ladder will remain above water, assuming that 3.5 metres was above water when the tide began to rise?


#14 Rare Books

In my collection of 80 rare books, 34 have leather covers and 18 were published before 1900. If 42 of these books were published after 1900 and do not have leather covers, how many books were published before 1900 and have leather covers?


#15 The Hikers

Four couples decide to hike together in the Snowy Mountains every summer. The men’s names are Zac, James, Brad and Tom. The women’s names are Bridie, Emma, Nikita and Susie. Use the following clues to determine each set of husbands and wives.

1. Nikita is Zac’s sister.hiking2

2. Susie has two brothers, but her husband is an only child.

3. Tom was best man at Nikita’s wedding.

4. The names of Bridie and her husband both begin with the same initials.

#16 abcd

Find a four-digit number abcd such that if a decimal point is placed between ab and cd, (i.e., the resulting number is the mean of ab and cd.


#17 Red Hats

Three women – A, B and C – are blindfolded and told that either a red or green hat will be placed on each of them. After this is done, the blindfolds are removed. The women are asked to raise a hand if they see a red hat, and to leave the room as soon as they are sure of the colour of their own hat.

All three hats happen to be red, so all three women raise a hand. Three minutes go by until C, who is more astute than the others, leaves the room.

How did she deduce the colour of her hat?


#18 Teddy Bears

Tom Tranter makes teddy bears in Tatura. He makes two sizes of teddy bears: a small bear that he sells for $2.50 and a larger bear that he sells for $5.75. Yesterday Tom made $367. Before he opened his shop in the morning, he had 200 teddy bears in his inventory. At the end of the day he had 126. How many bears of each price did he sell?



Solve the five simultaneous equations below to find the values of all five unknowns:

A+B+C+D = 16

B+C+D+E = 10

A+C+D+E =  8

A+B+D+E =  6

A+B+C+E =  4


Image from:  One Stop Card Crafts


#20 Exponents


Arrange the following in ascending order of size:

2 800 , 3 600 , 5 400 , 6 300


Image from:  Coolmath

#21 Triangle

The longest side of a triangle is 10 cm in length. If the shortest side is 3 cm in length, what can be said about the length of the third side?


#22 Brackets

What is the answer to:


Insert pairs of brackets into this expression to also get answers of 3, 4, 5 and 6.

#23 Aussie Rules

SherrinWhat scores in Australian Rules Football are such that the product of the number of goals and the number of behinds equals the actual points score? (note that in Aussie rules one goal equals six points, while one behind equals one point)

Example: 2 goals  12 behinds =  24 points;  and  2 x 12 = 24

#24 Missing Cubes

How many missing unit cubes need to be added to the diagrams below to make full cubes?


#25 Nine Dogs

There are 9 dogs in a square pen. By adding two more square pens, is it possible to ensure that each dog has a seperate enclosure?


Can you identify the 9 different dog breeds shown?

#26 TriMax

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

7 5
2 4 6
8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:


#27 The Present

How much red ribbon would be needed for the package shown below? If the present was given to you, what would you like it to contain?


#28 Darts

The drawing below shows a circle with a radius of r = 3 cm inside a circle with a radius of R = 5 cm. If a dart hits somewhere at random inside the larger circle, what is the probability that it will fall somewhere inside the smaller circle?


#29 Delicious Doughnuts




4 in a box



6 in a box



4 in a box



5 in a box

Top Cat’s footy team bought 11 boxes of delicious doughnuts to help celebrate their premiership victory. They bought 50 doughnuts in all.

  • They bought more boxes of jelly doughnuts than any other kind.
  • They bought the same number of boxes of original doughnuts as boxes of chocolate doughnuts.
  • They purchased more boxes of glazed doughnuts than original doughnuts.
  • They bought no more than 4 boxes of any one type of doughnut.

#30 Algebra Animals

What number does each animal represent, in order to make the row and column totals correct in the table below?

algebra puzzle

#31 Greatest Product

What is the greatest product of three adjacent numbers in any direction (up, down, left, right, or diagonally) in the 6 by 6 grid?


#32 Domino Squares

Take a double six set of dominos.  Arrange four dominoes in a square so that the sum of the numbers along each side is the same. Repeat this six more times so that all the dominoes are used and you have 7 correct squares. One possible square is shown in the diagram below.

domino square

#33 One Billion


#34 The Dalmation

How many spots are on the dalmation?


1.  The number of spots is divisible by 3

2.  When the number of spots is divided by the number of legs, a remainder of 3 results

3.  The spots can also be divided by the total of legs, ears, eyes and tail to leave a remainder of 6

#35 Counters


A total of 16 counters are put into four piles so that each pile has a different number of counters. List all possible arrangements of the counters. Write, and solve, a further question for investigation similar to this.

#36 Five Cards

Zoe had these five numbered cards:


How many five-digit numbers can she make that are divisible by 9 without having a remainder?

#37 Maths Textbook

Rachel accidentally tore some consecutive pages out of her Maths textbook. Rachel discovered that the sum of the page numbers was 261.


How many pages and what were the page numbers on the pages Rachel tore out of her book?

#38 Shade Six

Shade 6 circles on the diagram below, so that each row, column and diagonal has an even number of blank circles.


#39 Double Twelve Dominoes

Some double twelve dominoes are shown in the phot below. How many dominoes are there in a full double twelve set? How many dots do they have in total?



#40 Bomber Pilot

Suppose that you are a bomber pilot flying a B-17 during World War II. You know that 4% of bombers get shot down on average on each mission. Calculate the chance that you would successfully fly all the missions of your tour of duty, let’s say 50 (a larger number than was actually asked of air crews), without getting shot down.



#41 Cryptogram

Cryptograms are encoded sentences where each letter has been substituted for another. A letter may not stand for itself. Can you decode this cryptogram?


4 Responses to Problem Solving

  1. Michelle Norton says:

    #9 Apples for Sale
    You would have to sell 1500 apples to make a $100 profit. We worked it out by using the lowest common multiple of 3 and 5. Then we could work out the profit for 15 apples was $1. So we needed to sell 15 apples 100 times which is 1500 apples.

  2. theodore says:

    I enjoyed your problem solving section, but could not solve them all, could you send me some of the answers?

  3. Sarah says:

    These are great! Would you please be able to supply me with the answers? Id love to use some of these as challenges for my students. Thank you 🙂

  4. fnils-kmix says:

    If we find the prime factors of one billion ,we get:
    1000000000 = 2^9 * 5^9 = 512 * 1953125
    Therefore the two whole numbers that don’t end in zero and have a product of exactly 1000000000 are : 512 and 1953125

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