Maia the bee can only walk on coloured houses. In how many ways can you colour exactly three white houses, all the same colour, so that Maia can walk from A to B?
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Answer: B — 16
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Hint 1 of 2
You need a connected colored path linking A and B using exactly three newly-colored white houses.
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Hint 2 of 2
Count the distinct sets of three white houses that connect A to B.
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Approach: count valid 3-house connecting paths
Maia needs a connected run of colored houses from A to B.
Colour exactly three white houses so that, together with A and B, they form a connected walk.
Carefully listing the choices of three white houses that complete a path gives 16 ways.
Five sparrows are sitting on a rope (see picture). Some of them are looking to the left, some of them are looking to the right. Every sparrow whistles as many times as the number of sparrows it can see sitting in front of it. For example, the third sparrow whistles exactly twice. How many times do all the sparrows whistle altogether?
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Answer: D — 10
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Hint 1 of 2
Each sparrow only sees the birds in the direction its beak is pointing.
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Hint 2 of 2
Count how many birds are in front of each sparrow, then add all five counts.
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Approach: count each sparrow's forward view and add them up
Look at which way each beak points, then count the sparrows in front of it.
From left to right the sparrows look right, left, right, left, right, so they see 4, 1, 2, 3, and 0 birds in front.
How many numbers, which are only allowed to contain the digits 1, 2 or 3, are bigger than 10 and smaller than 32? The digits can be used more than once in the numbers.
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Answer: D — 7
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Hint 1 of 3
Bigger than 10 and smaller than 32 means the number has two digits and starts with 1, 2, or 3.
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Hint 2 of 3
Be neat: write all the numbers that start with 1, then all that start with 2, then those that start with 3.
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Hint 3 of 3
Remember each digit can only be 1, 2, or 3, and don't forget to stop before 32.
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Approach: list the two-digit numbers in order, using only the digits 1, 2, 3
We want two-digit numbers made only from 1, 2, 3 that are bigger than 10 and smaller than 32.
Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3 (but under 32): just 31.
Alex threads white and black beads alternately onto a piece of string. Twice, 5 beads are hidden — see picture. How many white beads are hidden in total?
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Answer: C — 6
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Hint 1 of 3
The beads always go white, black, white, black — never two of the same colour together.
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Hint 2 of 3
Look at the visible bead right before each hidden part to know what colour comes next.
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Hint 3 of 3
Fill in each hidden run of 5 by carrying on the colours, then count just the white ones.
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Approach: carry on the alternating colour pattern through each hidden run
The beads keep switching: white, black, white, black, and so on.
Each hidden group of 5 carries on that pattern, and in each one 3 of the 5 beads come out white.
There are two hidden groups, so 3 and 3 make 6 white beads in total. The answer is C.
The picture shows a mouse and a piece of cheese. The mouse is only allowed to move to the neighbouring fields in the direction of the arrows. How many paths are there from the mouse to the cheese?
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Answer: E — 6
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Hint 1 of 2
The arrows only let the mouse go forward toward the cheese, never back.
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Hint 2 of 2
Trace one path with your finger, then carefully find every different way without repeating one.
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Approach: trace every allowed path one at a time and count them
Put your finger on the mouse and follow the arrows toward the cheese.
Each time you reach a spot with two arrows, you can pick a different way to go.
Carefully trace each different route all the way to the cheese without repeating one.
Counting all the different routes gives 6, so the answer is E.
You make two-digit numbers using the digits 2, 0, 1 and 8. Each number must be bigger than 10 and smaller than 25, and made of two different digits. How many different numbers do you get?
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Answer: A — 4
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Hint 1 of 3
The number is between 10 and 25, so it must start with a 1 or a 2 — try each.
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Hint 2 of 3
For a number starting with 1, the other digit comes from 2, 0, 8 (a 1 would repeat).
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Hint 3 of 3
Write out every number you can make, then cross off any below 11 or 25 and up, and any with two equal digits.
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Approach: list every allowed number and count them
The number is bigger than 10 and smaller than 25, so it starts with 1 or 2.
Starting with 1, the other digit (from 2, 0, 8) gives 12, 10, 18 — but 10 is not bigger than 10, so keep 12 and 18.
Starting with 2, staying under 25, gives 20 and 21. All together: 12, 18, 20, 21, which is 4 numbers.
In which picture are there half as many circles as triangles and twice as many squares as triangles? (The five pictures are shown as choices A, B, C, D, E.)
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Answer: E
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Hint 1 of 2
For each picture, count the circles, the triangles, and the squares.
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Hint 2 of 2
You want the triangles to be in the middle: half as many circles, and double as many squares.
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Approach: count the shapes in each picture
Count the three kinds of shape in each picture.
We need a picture where the circles are the small group, the triangles are double the circles, and the squares are double the triangles.
For example 1 circle, 2 triangles, 4 squares fits: circles are half the triangles and squares are twice the triangles.
A hen lays white and brown eggs. Lisa takes six of them and puts them in a box as shown. The brown eggs are not allowed to touch each other. What is the largest number of brown eggs Lisa can put in the box?
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Answer: C — 3
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Hint 1 of 2
Two round eggs touch only when their cups are right next to each other (side by side or one above the other).
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Hint 2 of 2
Try putting brown eggs in cups that skip a space, like a checkerboard pattern.
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Approach: spread the brown eggs out so none are next to each other
The box has 6 cups in 2 rows of 3. Eggs touch only when their cups are side by side or one directly above the other.
Put brown eggs in the two top corners and the middle cup of the bottom row — none of these three cups touch.
A fourth brown egg would have to sit next to one of them, so the most Lisa can place is 3.
3 girls and 2 boys were dancing. They danced in pairs so that each girl danced with each boy for exactly 1 minute. At any time, there was only one pair on the dance floor. For how many minutes did they dance?
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Answer: B — 6
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Hint 1 of 3
Only one pair dances at a time, so the total minutes equals the number of pairs.
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Hint 2 of 3
Each girl needs to dance once with each boy.
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Hint 3 of 3
Count all the different girl-and-boy pairs you can make.
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Approach: count the pairs
There are 3 girls and 2 boys, giving 3 × 2 = 6 different pairs.
Each pair dances for 1 minute, one pair at a time.
Each participant in a cooking contest baked one tray of cookies like the one shown beside. What is the smallest number of trays of cookies needed to make the following plate?
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Answer: C — 3
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Hint 1 of 3
Count how many of each kind of cookie the big plate needs.
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Hint 2 of 3
Now see how many of each kind you get from just one tray.
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Hint 3 of 3
The kind of cookie you need the most of decides how many trays you must bake.
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Approach: compare each cookie type to what one tray gives
Count each kind of cookie on the plate, and count how many of that kind one tray makes.
For each kind, see how many trays it would take, then pick the biggest of those numbers.
Three trays are enough to cover every kind, so the answer is 3.
