Start with the number in the first cloud and follow the arrows. The arrows say, in order, −0, then +1, then ×5. Which number is hidden behind the question mark?
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Answer: E — 15
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Hint 1 of 2
Start with the number in the first cloud and do what each arrow says, one arrow at a time.
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Hint 2 of 2
Taking away 0 keeps the number the same; then add 1, then make 5 copies of it.
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Approach: follow the operation chain left to right
Begin with 2. The first arrow says −0, so it stays 2.
Florian has 10 identical metal strips, each with the same number of holes. He bolts them together in pairs to make the 5 long strips in the picture. Which of the long strips is the longest?
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Answer: A
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Hint 1 of 2
Each long strip is two short strips laid end to end, sharing a few holes where they overlap.
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Hint 2 of 2
The fewer holes the two short strips share, the further the long strip stretches.
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Approach: compare how much each pair of strips overlaps
Every long strip is two equal short strips bolted so they share some holes in the overlap.
Picture sharing only 1 hole versus sharing many: the less the strips overlap, the longer they reach.
Strip A is the one whose two pieces overlap the least, so it stretches the furthest.
The two shapes below each stand for a number. The red triangle plus 4 equals 7, and the blue square plus the red triangle equals 9. Which number is hidden behind the square?
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Answer: E — 6
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Hint 1 of 2
The first picture-sum tells you the triangle's number all by itself.
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Hint 2 of 2
Once you know the triangle, use the second picture-sum to find the square.
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Approach: find the triangle first, then use it to find the square
Triangle and 4 make 7, so the triangle must be 3 (because 3 + 4 = 7).
The square and the triangle make 9, so the square and 3 make 9.
The number that goes with 3 to make 9 is 6 (since 6 + 3 = 9).
Luis has got 7 apples and 2 bananas. He gives 2 apples to his friend Jacob, who gives him bananas in return. Afterwards Luis has got the same amount of apples as bananas. How many bananas did Luis get from Jacob?
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Answer: B — 3
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Hint 1 of 2
First update the apple count after Luis gives 2 away.
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Hint 2 of 2
Set the new banana count equal to the apple count and read off how many bananas he received.
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Approach: track apples and bananas, then make them equal
Luis gives away 2 of his 7 apples, leaving 5 apples.
Afterwards he has as many bananas as apples, so he must have 5 bananas.
He started with 2 bananas, so he received 5 − 2 = 3 bananas from Jacob.
Jack makes a cube from 27 small cubes. The small cubes are either grey or white as shown in the diagram. Two small cubes with the same colour are not allowed to be placed next to each other. How many small, white cubes has Jack used?
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Answer: C — 13
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Hint 1 of 2
No two cubes of the same colour may touch, so the colours alternate like the dark-and-light squares on a checkerboard.
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Hint 2 of 2
The big cube is built from 27 little cubes; the corners are grey, so count up the grey cubes and the rest are white.
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Approach: colour the 27 little cubes like a checkerboard
Since same-coloured cubes can't touch, the colours flip back and forth like a checkerboard going up, across and back.
Start the corner as grey: then the grey cubes are the 8 corners and the 6 little cubes sitting in the middle of each face — that is 8 + 6 = 14 grey cubes.
All 27 cubes minus the 14 grey ones leaves the white cubes: 27 − 14 = 13.
10 runners start in a running race. At the finish, there are 3 more runners behind Thomas than there are in front of him. In which position did Thomas finish?
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Answer: C — 4
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Hint 1 of 2
Besides Thomas there are 9 other runners, split into a front group and a back group.
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Hint 2 of 2
The back group is 3 bigger than the front group; try sharing the 9 runners so the back has 3 more.
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Approach: share the other 9 runners into front and back groups
Take Thomas out for a moment: the other 9 runners are split into the ones in front and the ones behind.
The back group must be 3 bigger than the front group, so split 9 into 3 in front and 6 behind (6 is 3 more than 3).
With 3 runners in front of him, Thomas is the next one, so he is in 4th place.
Joseph has got a toy car, a teddy bear, a ball and a ship. He wants to put them in a new order on the shelf. The ship must be next to the car, and the teddy bear should also be next to the car. In how many different orders can he put the toys on the shelf?
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Answer: B — 4
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Hint 1 of 2
The car must touch both the ship and the teddy, so the car sits between them.
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Hint 2 of 2
Treat ship-car-teddy as one block and place the ball at either end.
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Approach: form the forced block, then place the remaining toy
The ship and the teddy both must be next to the car, so the car is in the middle of a block: ship–car–teddy.
That block can be ordered 2 ways (ship–car–teddy or teddy–car–ship).
The ball goes at the left end or the right end of the block: 2 choices.
Peter rides his bike along a cycle path in a park. He starts at point S and rides in the direction of the arrow. At the first crossing he turns right, then at the next left, and then again to the right and then again to left. Which crossing does he not reach?
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Answer: D — D
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Hint 1 of 2
Put your finger on S, point it the way the arrow points, and ride along.
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Hint 2 of 2
At each crossing make the next turn in the list (right, then left, then right, then left) and see which labelled crossing your finger never lands on.
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Approach: trace the route obeying the turn sequence
Start at S facing the arrow and ride to the first crossing, then turn as told: right, then left, then right, then left.
Tracing this with your finger, Peter rolls through the other crossings but his turns steer him away from one of them.
Two of the 5 ladybirds in the picture are always friends with each other if the difference between their number of dots is exactly 1. Today every ladybird has sent an SMS to each of their friends. How many SMS messages were sent?
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Answer: C — 6
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Hint 1 of 2
First count the spots on each ladybird, then pair up those that differ by exactly one spot.
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Hint 2 of 2
Each friendship means two messages, since each friend texts the other.
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Approach: find friend pairs, then count messages both ways
The ladybirds carry 2, 3, 3, 5 and 6 spots.
Friends differ by exactly 1 spot: the 2-spot is friends with each 3-spot, and the 5-spot is friends with the 6-spot — 3 friendships in all.
In each friendship both friends send a message, so each pair accounts for 2 messages.
There are 10 balls, numbered 0 to 9 in a basket. John and George play a game. Each person is allowed to take three balls from the basket and calculate the total of the numbers on the balls. What is the biggest possible difference between the John and George's totals?
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Answer: E — 21
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Hint 1 of 2
To make the gap as large as possible, one player grabs the biggest numbers and the other the smallest.
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Hint 2 of 2
They draw from the same basket, so the two sets of three balls cannot overlap.
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Approach: maximise one total and minimise the other
One player can take the three largest balls: 9 + 8 + 7 = 24.
The other is then left to take the three smallest: 0 + 1 + 2 = 3.
Luca wants to cut the shape in figure 1 into equally sized small triangles (like those in figure 2). One of these triangles is already drawn on figure 1. How many of these triangles will he get?
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Answer: D — 15
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Hint 1 of 2
Notice that one little triangle is exactly half of a small grid square.
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Hint 2 of 2
If you know how much room the big shape covers in grid squares, two triangles fit in each square.
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Approach: fit the half-square triangles into the shape
Each little triangle is half of a small grid square, so two of them fill one square.
The big shape covers seven-and-a-half squares of room, and two triangles fit in every square.
Doubling seven-and-a-half gives 15 little triangles.
Some of the small squares on each of the square transparencies have been coloured black. If you slide the three transparencies on top of each other, without lifting them from the table, a new pattern can be seen. What is the maximum number of black squares which could be seen in the new pattern?
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Answer: D — 8
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Hint 1 of 2
Stacking the see-through sheets makes a square black wherever any sheet is black there.
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Hint 2 of 2
Slide them so the black cells overlap as little as possible, then count the covered squares.
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Approach: overlay the transparencies and count the black cells
Each sheet is transparent, so a cell looks black if it is black on at least one of the stacked sheets.
Lining the three sheets up so their black cells barely overlap covers as many squares as possible.
Together the black cells can cover 8 of the 9 squares, leaving just one clear.
The numbers 1, 2, 3, 4 and 9 are written into the squares on the following figure. The sum of the three numbers in the horizontal row should be the same as the sum of the three numbers in the vertical column. Which number is written in the middle?
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Answer: E — 9
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Hint 1 of 2
The middle square sits in both the row and the column, so it gets used in both totals.
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Hint 2 of 2
The other four numbers must split into two equal-sized pairs, one pair for the row's ends and one for the column's ends.
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Approach: the middle is shared, so the other four split into two equal pairs
The number in the middle is part of both the row total and the column total, so the other four numbers fill the two ends of the row and the two ends of the column.
Those four numbers must make two pairs that add up to the same amount; from 1, 2, 3, 4 the pairs 1 + 4 = 5 and 2 + 3 = 5 match perfectly, leaving 9 for the middle.
Then each line totals 9 + 5 = 14, the same both ways, so it works.
The middle number is 9.
A quick check for older kidsAll five numbers add to 19. The row and column together use the middle twice, giving 19 + middle, and that must split into two equal halves — so 19 + middle is even, which forces the middle to be odd, and 9 is the choice that balances.
In this square there are 9 dots. The distance between the points is always the same. You can draw a square by joining 4 points. How many different sizes can such squares have?
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Answer: D — 3
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Hint 1 of 2
Squares can sit straight on the dots, but they can also be tilted like a diamond.
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Hint 2 of 2
Hunt for a tiny straight square, a big straight square, and one slanted square.
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Approach: find every square size, straight ones and the tilted one
On the 3-by-3 dots you can make a tiny straight square (1 step on each side) and a big straight square (2 steps on each side).
You can also make a slanted square shaped like a diamond, joining the four middle dots of the edges.
Thomas drew a pig and a shark. He cuts each animal into three pieces. Then he takes one of the two heads, one of the two middle sections and one of the two tails and lays them together to make another animal. How many different animals can he make in this way?
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Answer: E — 8
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Hint 1 of 2
A new animal needs a head, a middle and a tail, and there are two choices for each part.
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Hint 2 of 2
Multiply the number of choices for the three parts.
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Approach: multiply the choices for each of the three parts
There are 2 heads to choose from, 2 middle sections, and 2 tails.
Each new animal is one choice for each part, so 2 × 2 × 2 = 8 animals can be built.
Anna, Berta, Charlie, David and Elisa baked biscuits at the weekend. Anna baked 24, Berta 25, Charlie 26, David 27 and Elisa 28 biscuits. By the end of the weekend one of the children had twice as many, one 3 times, one 4 times, one 5 times and one 6 times as many biscuits as on Saturday. Who baked the most biscuits on Saturday?
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Answer: C — Charlie
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Hint 1 of 2
Each child's weekend total is their Saturday pile shared into 2, 3, 4, 5 or 6 equal groups (each size used once).
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Hint 2 of 2
Find which totals can be shared evenly by 5 and by 3 first, since only one each can.
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Approach: see how each total splits evenly, then share it out to find Saturday
Each total (24, 25, 26, 27, 28) is a Saturday pile copied 2, 3, 4, 5 or 6 times, with each copy-number used once.
Only 25 shares evenly into 5 groups (25 ÷ 5 = 5) and only 27 shares evenly into 3 groups (27 ÷ 3 = 9).
That leaves 26 as 2 copies (26 ÷ 2 = 13), 28 as 4 copies (28 ÷ 4 = 7), and 24 as 6 copies (24 ÷ 6 = 4).
Saturday piles: Anna 4, Berta 5, Charlie 13, David 9, Elisa 7 — Charlie's 13 is the biggest, so the answer is Charlie.
