On every birthday Maria gets as many teddies as the age she turns: 1 teddy on her first birthday, 2 teddies on her second birthday, and so on. How many teddies does Maria have in total the day after her sixth birthday?
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Answer: C — 21
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Hint 1 of 2
On each birthday she gets a number of teddies equal to her age that day.
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Hint 2 of 2
Add up the teddies from birthday 1 through birthday 6.
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Approach: add the gifts from each birthday
Birthdays 1 to 6 give 1, 2, 3, 4, 5 and 6 teddies.
Add them up: 1 + 2 + 3 + 4 + 5 + 6 = 21.
So the day after her sixth birthday she has 21 teddies (choice C).
One of the five coins A, B, C, D or E should be moved to an empty square so that each row and each column ends up with exactly two coins. Which coin should be moved?
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Answer: C — C
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Hint 1 of 2
Count how many coins are in each row and in each column right now.
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Hint 2 of 2
Find the one row that has too many and the one column that has too many; the coin sitting where they cross is the one to move.
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Approach: balance rows and columns
Counting coins, one row has three coins (too many) and one row has only one (too few).
Likewise one column has three coins and another has only one.
The coin that sits in BOTH the overloaded row and the overloaded column is the one to move.
That coin is C; moving it to the empty cell of the short row and short column fixes every count to two.
When a laser beam hits a mirror it changes direction (see the small diagram). Each mirror reflects on both of its sides. At which letter does the laser beam come out?
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Answer: B — B
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Hint 1 of 3
Put your finger where the beam starts and slide it along, but turn a corner every time you reach a slanted mirror.
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Hint 2 of 3
A mirror leaning like '\' turns a beam going across into a beam going down; a mirror leaning like '/' turns it the other way.
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Hint 3 of 3
Keep sliding and turning until your finger walks off the edge at one of the letters.
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Approach: trace the beam one mirror at a time
Start your finger on the beam and slide it straight until it touches the first slanted mirror.
At each mirror, make a quarter turn the way the mirror leans, then keep sliding.
Following every bounce, the finger leaves the grid at the letter B.
Kengu hops to the right along the number line (see diagram). He makes one big jump and then two little jumps, and repeats this pattern again and again. He starts at 0 and lands on 16. How many jumps does Kengu make in total?
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Answer: E — 12
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Hint 1 of 3
Look at the picture: how many numbers does the big jump cover, and how many does each little jump cover?
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Hint 2 of 3
One round is big-little-little; work out how far one whole round moves him and how many jumps that is.
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Hint 3 of 3
Then skip-count by that round-distance until you land on 16.
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Approach: measure one round, then skip-count to 16
From the picture, the big jump moves 2 spaces and each little jump moves 1 space.
So one round (big, little, little) moves him 2 + 1 + 1 = 4 spaces using 3 jumps.
Skip-counting by 4 reaches 16 after 4 rounds (4, 8, 12, 16).
Logic & Word Problemswork-backwardcareful-counting
Five cars are labelled 1 to 5 and drive in the direction of the arrow. First the last car overtakes the two cars in front of it. Then the car that is now second to last overtakes the two in front of it. Finally the car that is now in the middle overtakes the two in front of it. In what order do the cars drive now?
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Answer: B — 2, 1, 3, 5, 4
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Hint 1 of 3
Write the five car numbers in a row (front car first) and act out the story move by move.
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Hint 2 of 3
When a car overtakes the two in front of it, slide it forward so it sits just ahead of both of those two cars.
Still stuck? Show hint 3 →
Hint 3 of 3
Do the three moves one at a time and read off the new order at the end.
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Approach: act out the overtakes one move at a time
The arrow points left, so the front-to-back order starts as 1, 2, 3, 4, 5 (car 1 leads).
The last car (5) jumps past the two in front of it (4 and 3): now 1, 2, 5, 3, 4.
The new second-to-last car (3) jumps past the two in front of it (5 and 2): now 1, 3, 2, 5, 4.
The car now in the middle (2) jumps past the two in front of it (3 and 1): now 2, 1, 3, 5, 4 — answer B.
Mosif has filled a table with numbers (see diagram). When he adds the numbers in each row and in each column, the result should always be the same, but he has made a mistake. To make every total the same he has to change one single number. Which number does Mosif have to change?
9
1
5
3
7
6
4
7
4
Show answer
Answer: B — 3
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Hint 1 of 2
Work out every row total and every column total and see which one is the odd one out.
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Hint 2 of 2
The number to change sits where the wrong row crosses the wrong column.
Show solution
Approach: find the row and column that are off
The row sums are 15, 16, 15 and the column sums are 16, 15, 15, so the target is 15.
One row is 1 too big and one column is 1 too big.
The cell in both that row and that column is the 3; lowering it to 2 fixes both.
Aladdin’s carpet is a square. Along each edge there are two rows of dots (see diagram), and each edge has the same number of dots. How many dots does the carpet have in total?
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Answer: A — 32
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Hint 1 of 3
The dots make two square loops — a big loop on the outside and a smaller loop just inside it.
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Hint 2 of 3
Count the dots on one side of a loop, but be careful: the corner dots belong to two sides, so don't count them twice.
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Hint 3 of 3
Add up the big loop and the small loop.
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Approach: count the big square loop and the small square loop of dots
The dots make two square loops, one just inside the other, with the same number on every side.
The big loop has 6 dots along each side; counting around it (corners only once) gives 4 × 6 − 4 = 20 dots.
The small loop has 4 dots along each side, giving 4 × 4 − 4 = 12 dots.
In a classroom the children sit in rows, with the same number of children in each row. In Robert’s row there are 2 children to his left and 3 children to his right. There are 2 rows in front of Robert and just 1 row behind him. How many children are in the class in total?
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Answer: E — 24
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Hint 1 of 2
Count the children in Robert's row including Robert himself.
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Hint 2 of 2
Then count the rows, again including Robert's own row.
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Approach: count one row and the number of rows, including Robert
In Robert's row there are 2 to his left, Robert, and 3 to his right: 2+1+3 = 6 children per row.
There are 2 rows in front, Robert's row, and 1 behind: 2+1+1 = 4 rows.
Johanna takes a paper with the numbers 1 to 36 and folds it in half twice (see diagrams). Then she pokes a hole through all four layers at once (see the diagram on the right). Which four numbers does she pierce?
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Answer: C — 14, 17, 20, 23
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Hint 1 of 2
Each fold lays one half exactly onto the other, so the hole goes through matching squares.
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Hint 2 of 2
Track which four numbers stack on top of each other at the hole's position.
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Approach: undo the folds to find the stacked numbers
The horizontal fold pairs each top-half square with the bottom-half square it lands on.
The vertical fold then pairs left columns with right columns.
The hole's spot stacks the squares 14, 17, 20 and 23.
Three football teams play in a tournament, and each team plays every other team once. A win is worth 3 points and a loss 0 points; a draw gives each team 1 point. Which number of points is impossible for any team to finish with?
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Answer: D — 5
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Hint 1 of 2
Each team plays only two games, scoring 3, 1 or 0 in each.
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Hint 2 of 2
List the totals you can build from two of {0, 1, 3} and see which option is missing.
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Approach: list every possible two-game total
Per game a team gets 3 (win), 1 (draw) or 0 (loss).
Wanda chooses some of the shapes shown. She says: “I have chosen exactly 2 grey, 2 big and 2 round shapes.” What is the smallest number of shapes Wanda could have chosen?
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Answer: B — 3
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Hint 1 of 2
You want a shape to count toward more than one of the requirements at once.
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Hint 2 of 2
Pick shapes that are grey-and-big, big-and-round, or grey-and-round to overlap the three needs.
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Approach: make each shape cover two requirements
She needs exactly 2 grey, 2 big and 2 round.
Choose a big grey shape (grey+big), a big round shape (big+round) and a small grey round shape (grey+round).
These three give exactly 2 grey, 2 big and 2 round.
A pyramid is built from cubes (see diagram), and every cube has side length 10 cm. An ant crawls along the line drawn across the pyramid (see diagram). How long is the path the ant takes?
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Answer: E — 90 cm
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Hint 1 of 3
The drawn line is made of short straight pieces, and each piece is exactly one cube-edge long.
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Hint 2 of 3
One cube edge is 10 cm, so you only need to count how many cube-edges the whole line covers.
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Hint 3 of 3
Trace the line up the steps and back down, counting one edge at a time.
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Approach: count the cube-edges the line covers, each 10 cm
The ant's line follows the steps of the pyramid, and every little piece is one cube-edge of 10 cm.
Tracing the line up over the steps and down the other side, it covers 9 cube-edges.
A road leads away from each of the six houses (see diagram), but the hexagon of roads for the middle is missing. Which hexagons can go in the middle so that you can travel from A to B and to E, but not to D?
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Answer: C — 1 and 5
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Hint 1 of 3
The roads inside the hexagon decide which houses get joined to which — put your finger on A and see where you can drive.
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Hint 2 of 3
You want A, B and E all on one set of connected roads, but D left out with no way to reach it.
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Hint 3 of 3
Try each hexagon in the gap and trace the roads from A every time.
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Approach: drop in each hexagon and trace the roads from A
Fit a hexagon into the gap, then put your finger on house A and follow every road you can drive along.
You need A, B and E to all join up, while D stays cut off (no road reaches it).
Only hexagons 1 and 5 connect A to B and E while leaving D alone.
Ahmed and Sara start at point A and walk in the directions shown, at the same speed. Ahmed walks around the square garden and Sara walks around the rectangular garden. How many rounds must Ahmed walk to meet Sara at point A again for the first time?
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Answer: C — 3
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Hint 1 of 3
Work out how far one lap is for each child by adding up the sides of their garden.
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Hint 2 of 3
Each child is back at A after 1 lap, 2 laps, 3 laps… so skip-count the total distance for each.
Still stuck? Show hint 3 →
Hint 3 of 3
Look for the first distance that shows up in both lists — that is when they meet at A.
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Approach: skip-count each child's distances until they match
Ahmed's square garden is 5 + 5 + 5 + 5 = 20 m around; Sara's rectangle is 10 + 5 + 10 + 5 = 30 m around.
Ahmed is back at A after 20, 40, 60… metres; Sara is back at A after 30, 60, 90… metres.
The first distance in both lists is 60 m, so that is when they meet at A again.
Ahmed has gone 60 ÷ 20 = 3 laps, so he walks 3 rounds (choice C).
Five girls eat plums. Laura eats 2 more plums than Sophie. Bettina eats 3 fewer plums than Laura. Clara eats one more plum than Bettina, and 3 fewer than Alice. Which two girls eat the same number of plums?
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Answer: E — Clara and Sophie
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Hint 1 of 3
Pretend Sophie eats some easy number of plums, like 10, then work out everyone else from the clues.
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Hint 2 of 3
Go in order: Laura is 2 more than Sophie, Bettina is 3 less than Laura, Clara is 1 more than Bettina.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you have all five numbers, look for two girls with the same count.
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Approach: pretend Sophie's number, then fill in the rest
Say Sophie eats 10 plums (any number works the same way).
Laura eats 2 more, so 12; Bettina eats 3 less than Laura, so 9; Clara eats 1 more than Bettina, so 10; Alice eats 3 more than Clara, so 13.
Now compare: Sophie has 10 and Clara has 10 — they match!
So the two girls who eat the same are Clara and Sophie (choice E).
For older kids (with letters)Let Sophie = S. Then Laura = S+2, Bettina = S−1, Clara = S, Alice = S+3, so Clara = Sophie.
The big cube is built from three different kinds of building blocks (see diagram). How many of the little white cubes are needed to build the big cube?
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Answer: B — 11
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Hint 1 of 3
First figure out how many little cubes fill the whole big cube — it is 3 across, 3 deep and 3 tall.
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Hint 2 of 3
Each grey L-piece and each dark bar is made of 3 little cubes, so count how many little cubes all the coloured pieces use up.
Still stuck? Show hint 3 →
Hint 3 of 3
Whatever little cubes are left over after the coloured pieces must be the single white ones.
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Approach: count all the little cubes, then take away the coloured pieces
The big cube is 3 across, 3 deep and 3 tall, so it holds 3 × 3 × 3 = 27 little cubes.
Each grey L-piece and each dark bar is built from 3 little cubes, and together the coloured pieces fill 16 of the 27 spots.
Every spot that is left over must be a single white cube: 27 − 16 = 11.
Cards of the same colour always hide the same number. When the three hidden numbers in a row are added, you get the number written to the right of that row (see diagram). Which number is hidden under the black card?
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Answer: D — 12
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Hint 1 of 3
Every card of the same colour hides the same number, so think of one secret number per colour.
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Hint 2 of 3
Compare two rows that are almost the same to find one colour's number first.
Still stuck? Show hint 3 →
Hint 3 of 3
Once you know grey and white, the black card is just its row total take away the other two.
Show solution
Approach: compare rows to peel off one colour at a time
The top row is grey + white + white = 34, and the bottom row is white + grey + grey = 26.
Both rows use one extra grey instead of one extra white, and they differ by 34 − 26 = 8, so a white card is 8 more than a grey card.
Trying small numbers that fit grey + 2 whites = 34: grey = 6 and white = 14 works (6 + 14 + 14 = 34).
The middle row grey + white + black = 32, so black = 32 − 6 − 14 = 12 — answer D.
For older kids (with letters)Let grey = g, white = w, black = k. From g + 2w = 34 and 2g + w = 26 you get g = 6, w = 14, then k = 32 − g − w = 12.
