Challenges

Puzzles, brain teasers and maths riddles are a fun way to use your problem-solving skills and logical thinking.

Sharpen those maths skills and have a go at this fun challenge.

How many triangles?

How many triangles can you find in this pattern?

A six-pointed star divided into multiple smaller triangles by intersecting lines. Each point of the star contains a smaller triangle, and the interior of the star is divided into triangles of the same size. These can be combined to create triangles of various sizes.

There are 20 triangles in total. Did you find them all?

The text ‘12+6+2 = 20 triangles’ is at the top above an image of three stars on a yellow background. The first star, labelled ‘x12’, is a six-pointed star divided into 12 smaller triangles by intersecting lines. One of the small triangles in a star point is shaded red to highlight it Next to the first star is another six-pointed star, labelled ‘x6’. It is also divided into 12 smaller triangles by intersecting lines. A larger triangle, made by combining 4 of these smaller triangles, is shaded red to highlight it. Below both stars is a third six-pointed star, labelled ‘x2’. It is also divided into 12 smaller triangles by intersecting lines. An even larger triangle, made by combining 9 of these smaller triangles, is shaded red to highlight it.

Missing number

Can you find the missing number? Which number should go in the empty triangle?

Four triangles are arranged on a yellow background. The first triangle is blue, and the number 4 is written inside. On each point of the triangle is a number: 8, 3 and 4. Next to that triangle is an orange triangle with the number 3 written inside. On each point of the triangle is a number: 6, 5 and 13. Below these two triangles are two more triangles. One triangle is green and has the number 7 written inside. On each point of the triangle is a number: 2, 5 and 9. Next to the green triangle is a light blue triangle with a question mark written inside. On each point of the triangle is a number: 4, 6 and 14.

Did you find the pattern? Add the bottom two numbers and divide it by the top number to find the number in the middle. Therefore, the pattern to find the missing number is: (14 + 6) ÷ 4, which means the missing number is 5.

A light blue triangle on a yellow background has the number 5 written inside. On each point of the triangle is a number: 4, 6 and 14.

Fruitful

Sam has three boxes of fruit. One contains pears, one contains oranges, and one contains both oranges and pears.

The labels have fallen off and all have been stuck back on the wrong boxes.

Sam opens one box and, without looking in the box, takes out one piece of fruit. Sam looks at the fruit, the label on the box and immediately put all the labels on the correct boxes.

What label was on the box Sam opened and what fruit did they take out?

Three labelled boxes on yellow background. One is labelled ‘Pears’, another is labelled, ‘Oranges’ and the third is labelled ‘Pears and Oranges’.

There are two possible solutions, here is one:

Sam opened the box labelled ‘Pears & Oranges’. As all labels are wrong, this box must contain only pears or only oranges. If they pick a pear, then they know that the box they opened should be labelled ‘Pears’ and other two should be ‘Oranges’ and ‘Pears & Oranges’. The box labelled ‘Oranges’ should be labelled as ‘Pears & Oranges’ and the box labelled ‘Pears’ should be labelled ‘Oranges’.

Confused? Take a look at this table.

A table on a yellow background. The table is divided into three rows of four columns. The top row is blue and the first column is the text ‘Sam picks’, the next column has text ‘Pears’, the next has text ‘Oranges’ and the final column has the text ‘Pears & Oranges’. On the next row underneath ‘Sam picks’ is the text “A pear’, in the next column is the text ‘Oranges’, the next column has the text ‘Pears & Oranges’ and the final column has the text ‘Pears’. Along the bottom row, the first column has the text ‘An orange’, the next column has ‘Pears & Oranges’, next is text ‘Pears’ and in final column is the text ‘Oranges’.

Four lines 

Draw 9 dots in a square like this: 

Nine black dots arranged equally spaced as a three by three grid on yellow background.

Can you draw a line through all 9 dots with just four straight lines?

You can start wherever you like but you can’t take your pen off the paper.

For this one, you must (literally) think outside the box! 

Here is one solution:

Nine black dots arranged equally spaced as a three by three grid on yellow background. A blue line is drawn to represent four lines drawn to connect all the black dots without lifting the pen. An arrow drawn diagonally starts from the dot in the top left hand corner, passing through the middle dot and ending at the dot on the bottom right corner. The next line is drawn from this dot straight up passing through the dot on the right side extending beyond the dot in the top right corner. Then the next line is drawn at a 45 degree angle through the middle top of the top line and the middle dot of the left hand side extending beyond the dot in line with the bottom row of dots. The final line is down from this point along the bottom three dots finishing at the dot in the bottom right corner.

How many squares?

How many squares can you see in this pattern?

A white grid divided into small squares arranged in an array 7x6 layout (7 rows and 6 columns) on a yellow background. Along the top row three squares are missing after the first two squares. Along the left hand side two squares are missing leaving the first two and the bottom two squares. Along the bottom row two squares followed by three missing squares then two squares. Along the right hand side two squares are missing leaving the first two and the bottom two squares.

There are 56 squares in total: 32 at 1 x 1, 16 at 2 x 2, 6 at 3 x 3, 2 at 4 x 4.

At the top of the image the text ’32 + 16 + 6 + 2 = 56 squares’ appears above 4 grid shapes on yellow background. The following grid shape is displayed four times: ‘A white grid divided into small squares arranged in an array 7x6 layout (7 rows and 6 columns) on a yellow background. Along the top row three squares are missing after the first two squares. Along the left hand side two squares are missing leaving the first two and the bottom two squares. Along the bottom row two squares followed by three missing squares then two squares. Along the right hand side two squares are missing leaving the first two and the bottom two squares. The four grids are labelled and shaded as follows: The first grid, labelled ‘x32’, has one small square shaded red, the second grid, labelled ‘x16’, has four small squares shaded red to represent a 2x2 square, the third grid, labelled ‘x6’, has nine small squares shaded red to represent a 3x3 square, the fourth grid, labelled ‘x2’, has sixteen small squares shaded red to represent a 4x4 square.

Number lines

Can you put the numbers 1 to 7 in each circle so that the total number of every line is 12?

A series of seven connected circles are arranged on a yellow background. The top row consists of three circles arranged equidistant in a horizontal line. Below this row, a single circle is placed in the middle, aligned with the centre of the row above. Below the middle circle is another row of three circles, also arranged equidistant in a horizontal line.

How did you go? Here is one solution. Teachers, if students get stuck on this one, give them the middle number 4 as a hint.

A series of seven connected circles are arranged on a yellow background. The top row consists of three circles arranged equidistant in a horizontal line. Each circle has a number inside: 7, 2 and 3. Below this row, a single circle with the number 4 inside is placed in the middle, aligned with the centre of the row above. Below the middle circle is another row of three circles, also arranged equidistant in a horizontal line. Each of these circles has a number inside: 5, 6 and 1.

The escaping frog

A frog has fallen into a pit that is 30 metres deep. Each day the frog climbs up 3 metres but falls back 2 metres at night.

How many days does it take for the frog to escape?

A line drawing of a pit labelled 30 m in depth, with the height marked by an arrow. A small amount of water is drawn at the bottom of the pit. A frog is shown attempting to climb the wall of the pit. At the top of the pit on both sides, there are grassy strips drawn to indicate ground level.

It will take 28 days for the frog to escape. After 27 days and nights the frog only has 3 metres to go. On the 28th day the frog is able to jump to freedom!

A line drawing of a pit labelled 30 m in depth, with the height marked by an arrow. A small amount of water is drawn at the bottom of the pit. At the top of the pit on both sides, there are grassy strips drawn to indicate ground level. On the grassy strip on the left hand side sits a frog.

Missing matches

Remove just 4 matches to leave 4 equilateral triangles. Equilateral means all the triangles must be exactly the same size.

Try using real matches or toothpicks to help work through the problem.

Sixteen brown matchsticks with red ‘heads’ are arranged in a pattern on yellow background. The central pattern represents a hexagon with six matchsticks radiating from a central point and a matchstick for each side. On the top side of the hexagon two matchsticks are added to form a triangular shape. On the bottom side of the hexagon two matchsticks are added to form another triangular shape.

Here is one solution. How many different solutions can you find?

Sixteen are arranged in a pattern on yellow background. The central pattern represents a hexagon with six matchsticks radiating from a central point and a matchstick for each side. On the top side of the hexagon two matchsticks are added to form a triangular shape. On the bottom side of the hexagon two matchsticks are added to form another triangular shape. Two of the matchsticks radiating from the central point are white. Two of the matches that make up the sides of the hexagon shape are also white. The remaining matchsticks are brown with red ‘heads’

Find the path

Can you find a path adding the numbers as you go to make exactly 53?

Start at the bottom left square (5) and move up, down, left or right until you reach the finish square (4).

 

A 5x5 grid is on a yellow background. Each grid square contains a number. On the bottom row, the first square on the lefthand side, is shaded green, contains the number five and is labelled ‘Start’ with an arrow. The following numbers on the squares are: 5, 6, 5, 5. On next row above, has squares with numbers: 7, 8, 8, 8, 6. On next row above, has squares with numbers: 6, 6, 4, 9, 9. On next row above, has squares with numbers: 8, 9, 4, 5, 7. On the top row the numbers in squares are: 4, 9, 7, 7. With the last square shaded in green with the number 4 and labelled ‘Finish’ with an arrow.

How did you go? Here is one solution:

5 + 7 + 6 + 6 + 9 + 4 + 5 + 7 + 4 = 53

 

A 5x5 grid is on a yellow background. Each grid square contains a number. On the bottom row, the first square on the lefthand side, is shaded green, contains the number five and is labelled ‘Start’ with an arrow. The following numbers on the squares are: 5, 6, 5, 5. On next row above, has squares with numbers: 7, 8, 8, 8, 6. The squares containing 9,4,5,7 are shaded in blue. On next row above, has squares with numbers: 6, 6, 4, 9, 9. The squares containing 6 are shaded in blue. On next row above, has squares with numbers: 8, 9, 4, 5, 7. The square containing 7 is shaded in blue. On the top row the numbers in squares are: 4, 9, 7, 7. With the last square also shaded in blue with the number 4 and labelled ‘Finish’ with an arrow.

Hidden shapes

What is the total number of triangles and the total number of squares in the diagram? Look carefully, some may be hidden.

A large 2x2 square grid is on yellow background. Each of the four squares in the grid is divided by two diagonal lines that intersect, forming four triangles within each square. In each square, two triangles are shaded green, and two triangles are shaded blue. The shading alternates within each square to create a symmetrical colour pattern across the entire grid.

The text ‘Total number of triangles’ appears at the top of the page above four squares arranged in a row. Each of the four squares is divided by a horizontal and a vertical line that intersect at the centre, forming four squares. Each square is divided by two diagonal lines that intersect, forming four triangles within each square. The first square has a triangle shaded blue and labelled ‘+16’. In the bottom corner two of the square, two adjacent small triangles are also shaded in blue to form a larger triangle. It is labelled ‘+16’. The second square has four adjacent small triangles shaded in blue, forming a larger triangle with one side equal to the side of the square. It is labelled '+4'. The third square has four adjacent small triangles shaded in blue, forming a larger triangle with one side equal to the vertical line that divides the square. It is labelled '+4'. The fourth square has eight adjacent small triangles shaded in blue, forming a larger triangle with one side equal to the side of the square and another side equal to the other side of the square. It is labelled '+4'. After the row of squares the text ‘=44’ is in a black circle to represent the answer to the total number of triangles. The text ‘Total number of squares’ appears under the row of squares. Below the text are another row of four squares arranged in a row. Each of the four squares is divided by a horizontal and a vertical line that intersect at the centre, forming four squares. Each square is divided by two diagonal lines that intersect, forming four triangles within each square. The first square has four adjacent triangles shaded blue to form a square. It is labelled ‘+14’. The second square has two small triangles shaded in blue, forming a smaller square. It is labelled '+4'. The third square has eight adjacent small triangles shaded in blue, forming a large square. It is labelled '+1'. The fourth square is fully shaded in blue. It is labelled '+1'. After the row of squares the text ‘=10’ is in a black circle to represent the answer to the total number of squares.