Bruno builds a big triangle out of small triangles that are all the same size. Some are already placed (shown grey). How many more small triangles does he need so that the big triangle is completely filled?
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Answer: B — 6
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Hint 1 of 3
The grey little triangles are already glued in; the white spaces are the holes still to fill.
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Hint 2 of 3
We only need to count the white triangles, because each hole needs exactly one more small triangle.
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Hint 3 of 3
Carefully point to and count every white triangle, one by one.
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Approach: count the white holes
The grey triangles are already placed, so we just need to fill the white holes.
Touch and count each white triangle one at a time.
There are 6 white triangles, so Bruno needs 6 more.
Tim has black and white squares of paper. He sticks the squares on the inside of a window so that the shown pattern appears. Which pattern can be seen from outside the window?
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Answer: D
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Hint 1 of 3
Think about looking at writing through glass from the other side — it comes out backwards.
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Hint 2 of 3
Seen from outside, the whole pattern is flipped like a mirror: left and right swap.
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Hint 3 of 3
Flip each row so the left square becomes the right square, then find the matching picture.
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Approach: flip the pattern left-to-right like a mirror
Looking from outside is like seeing the window through a mirror, so the picture flips left-to-right.
Take each row and swap its left and right squares: the black square on the left jumps to the right, and so on.
Logic & Word Problemscareful-countingcomplementary-counting
Chen has these 5 baskets, with 4 toys in each (shown as A, B, C, D, E in the picture). Four of the baskets fall down and the toys lie mixed up on the floor. Which basket did he not drop?
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Answer: B — B
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Hint 1 of 3
Look at the floor pile and notice which kind of toy is the most common there.
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Hint 2 of 3
The basket Chen kept is the only one with none of a toy that the others all had.
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Hint 3 of 3
Count the ducks: every duck is on the floor, so the kept basket has no duck.
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Approach: find the toy that is missing from the basket that stayed up
On the floor we can find ducks, frogs, ladybugs, and hippos all mixed together.
Count the yellow ducks on the floor: there are 6, which is every duck from the five baskets.
So the basket that did NOT fall has no duck in it — and the only basket with no duck is basket B.
In the table, each shape stands for a different number. The number at the end of each row is the sum of that row, and the number under each column is the sum of that column. What number does the star stand for?
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Answer: C — 3
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Hint 1 of 3
Each shape always means the same number, and a line of shapes adds up to the number at its end.
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Hint 2 of 3
Start with a line that has two of the SAME shape — that makes the shape easy to figure out.
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Hint 3 of 3
Once you know the heart, look at the middle column to find the star.
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Approach: figure out the easy shapes first, then the star
The left column is two smileys adding to 10, so each smiley is 5 (because 5 and 5 make 10).
The right column is two hearts adding to 4, so each heart is 2 (because 2 and 2 make 4).
The middle column is a heart and a star adding to 5; the heart is 2, and 2 plus 3 makes 5, so the star is 3.
Marina draws five pictures in a fixed order (a cloud, a ghost, a cat, a moon, and a flame), then repeats the same five over and over again. Which picture is the 27th picture?
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Answer: B
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Hint 1 of 3
The pictures come in repeating groups of five, always in the same order.
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Hint 2 of 3
Count by fives to get close to 27 without drawing them all out.
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Hint 3 of 3
After five whole groups you have drawn 25 pictures, so just keep going from there.
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Approach: count in groups of five
The five pictures repeat as a group, so skip-count: 5, 10, 15, 20, 25 finishes five whole groups.
Picture 25 is the last one of a group (the flame), so picture 26 starts a new group with the 1st picture (the cloud).
Picture 27 is then the 2nd picture of the group — the ghost, option B.
Anna, Bella, Che and Dimitry each have three shapes. Each child shares exactly one of their shapes with one other child. Anna has a triangle, a circle and a square; Bella has a heart, a square and a star; Che has a star, a triangle and a diamond. Which three shapes does Dimitry have?
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Answer: E
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Hint 1 of 3
Write out the shapes the other three children have and see which ones already come in matching pairs.
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Hint 2 of 3
Some shapes already appear for two children (a pair), but a few shapes appear for only one child so far.
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Hint 3 of 3
Dimitry must hold exactly the shapes that still need a partner, so that every shape ends up shared by two children.
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Approach: give Dimitry the shapes that still need a matching partner
Among Anna, Bella and Che, the square (Anna & Bella), the star (Bella & Che) and the triangle (Anna & Che) already come in pairs.
That leaves the circle (only Anna), the heart (only Bella) and the diamond (only Che) without a partner.
So Dimitry must have the circle, the heart and the diamond — this pairs every shape with exactly one other child, which is option E.
Zoran builds towers from three different building blocks (a triangle top, a rectangle, and an hourglass). The picture shows the heights of three towers. How high is the fourth tower?
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Answer: A — 12
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Hint 1 of 3
Each kind of block is always the same height, so the same block is worth the same number every time.
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Hint 2 of 3
Compare two towers that are almost the same — the difference in their heights tells you how tall the extra block is.
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Hint 3 of 3
Find the rectangle and the triangle heights, since the fourth tower is just those two stacked.
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Approach: compare towers to find each block's height
The tall tower (triangle + rectangle + hourglass) is 20, and the short one with just (triangle + hourglass) is 13; the only extra block is the rectangle, so the rectangle is 20 − 13 = 7.
The tower with (rectangle + hourglass) is 15, and the rectangle is 7, so the hourglass is 8; then in the tower of 13 the triangle is 13 − 8 = 5.
The fourth tower is just triangle + rectangle = 5 + 7 = 12.
Andrew throws arrows at a target. He starts with 10 arrows. Each time he hits the target, he gets 2 more arrows. In total Andrew throws 20 arrows, and then he has run out of arrows. How many times did Andrew hit the target?
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Answer: B — 5
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Hint 1 of 3
He only started with 10 arrows, but he threw 20, so where did the extra arrows come from?
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Hint 2 of 3
Every hit is like a little gift of 2 more arrows.
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Hint 3 of 3
Figure out how many extra arrows he earned, then see how many hits that took.
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Approach: count the extra arrows the hits gave him
He started with 10 arrows but threw 20 in total, so 10 extra arrows must have come from hitting the target.
Each hit gives 2 extra arrows, so we count by twos: 2, 4, 6, 8, 10 — that takes 5 hits to reach 10 extra arrows.
The two pictures show the same bridge at different times. All the cars are the same length. The numbers give the distances between the cars, and between a car and the end of the bridge. How long is each car?
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Answer: C — 5 metres
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Hint 1 of 3
It is the same bridge in both pictures, so the bridge is exactly the same total length both times.
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Hint 2 of 3
In each picture add up all the gap numbers, and count how many cars there are.
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Hint 3 of 3
The bottom picture has one fewer car but more gap — that extra gap must be exactly one car long.
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Approach: the same bridge length both times
Top picture: the gaps add to 1 + 2 + 1 + 2 = 6 metres, and there are 3 cars.
Bottom picture: the gaps add to 4 + 4 + 3 = 11 metres, and there are 2 cars; the bridge is the same length, so the bottom has 11 − 6 = 5 more metres of gap but exactly one fewer car.
That missing car is filling those extra 5 metres of gap, so each car is 5 metres long.
