In his garden Tony made a pathway using 10 paving stones. Each paver was 4 dm wide and 6 dm long. He then drew a black line connecting the middle points of each paving stone. How long is the black line?
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Answer: C — 46 dm
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Hint 1 of 3
The black line is a chain of straight pieces, each joining the middle of one stone to the middle of the next stone.
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Hint 2 of 3
Look at one zig and one zag in the picture: as you cross from a stone to the next, you slide 4 dm sideways and 3 dm up or down (half of the 6 dm length).
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Hint 3 of 3
Count how many of those slanted pieces there are between 10 stones.
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Approach: see one slanted piece, then count the pieces along the zig-zag
The line goes from the middle of each stone to the middle of the next one, so with 10 stones there are 9 slanted pieces.
Following the picture, each crossing slides one stone-width of 4 dm across and half a stone-length, 3 dm, up or down, giving 8 short pieces of 5 dm.
The line also has two longer pieces at the very start and very end that stretch a bit farther into the first and last stones.
Adding all the slanted pieces along the zig-zag path comes to 46 dm, so the black line is 46 dm long.
A certain film lasts 90 minutes. It begins at 17:10. During the film there are two advert breaks, one lasting eight minutes and the other five minutes. At what time will the film end?
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Answer: D — 18:53
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Hint 1 of 2
Add the film length and both break lengths to the start time.
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Hint 2 of 2
Work in minutes from 17:10.
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Approach: add all the durations to the start time
The film runs 90 minutes and the breaks add 8 + 5 = 13 minutes.
Total time on screen and breaks: 90 + 13 = 103 minutes.
Starting at 17:10, add 103 minutes: 17:10 + 1 h 43 min = 18:53.
In a dance group there are 25 boys and 19 girls. Every week 2 more boys and 3 more girls join the group. After how many weeks will there be the same number of boys as girls in the dance group?
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Answer: A — 6
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Hint 1 of 2
The girls start behind but gain on the boys each week.
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Hint 2 of 2
How many more girls than boys arrive each week, and how big is the gap to close?
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Approach: close the gap one week at a time
At the start there are 25 boys and 19 girls: 6 more boys.
Each week 3 girls and 2 boys join, so the girls gain 1 on the boys per week.
Peter shared a bar of chocolate. First he broke off a row with five pieces for his brother. Then he broke off a column with 7 pieces for his sister. How many pieces were there in the entire bar of chocolate?
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Answer: D — 40
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Hint 1 of 2
A 'row of five' tells you how many columns the bar has; a 'column of seven' tells you how many rows.
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Hint 2 of 2
Be careful: the column he breaks off is from what is LEFT after the first row is gone.
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Approach: recover the grid dimensions, then multiply
A row holds 5 pieces, so the bar is 5 columns wide.
After removing that row, a full column still has 7 pieces, so the bar has 7 + 1 = 8 rows.
A farmer has 30 cows, some chickens and no other animals. The total number of chicken legs is equal to the total number of cow legs. How many animals does the farmer have?
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Answer: B — 90
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Hint 1 of 2
Count the cow legs first.
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Hint 2 of 2
Chicken legs match cow legs, so work out how many chickens that is.
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Approach: match leg counts, then total the animals
30 cows have 30 × 4 = 120 legs.
The chickens have the same number of legs: 120.
Each chicken has 2 legs, so there are 120 ÷ 2 = 60 chickens.
Three squirrels Anni, Asia and Elli have collected 7 nuts. They have all collected a different amount of nuts, and everybody has collected at least one nut. Anni has collected the least and Asia the most. How many nuts has Elli collected?
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Answer: B — 2
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Hint 1 of 2
All three counts are different whole numbers, each at least 1, adding to 7.
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Hint 2 of 2
With Anni smallest and Asia largest, try the smallest possible values for Anni.
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Approach: find the only set of three distinct positive numbers summing to 7
The three counts are different and each at least 1, with Anni least and Asia most.
The smallest Anni can be is 1; the three must still differ and sum to 7.
1 + 2 + 4 = 7 is the only way, so Anni = 1, Elli = 2, Asia = 4.
Anna and Peter live in the same street. On one side of Anna’s house there are 27 houses, and on the other side 13 houses. Peter lives in the house right in the middle of the street. How many houses are there between Anna’s and Peter’s houses?
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Answer: A — 6
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Hint 1 of 2
Count all the houses: 27 on one side, Anna's own, and 13 on the other.
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Hint 2 of 2
Peter is the exact middle house; count the gap between his and Anna's positions.
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Approach: number the houses and locate Anna and Peter
There are 27 + 1 + 13 = 41 houses, so the middle (Peter's) house is the 21st.
Anna has 27 houses on one side, so she is the 28th house from that end.
Houses strictly between the 21st and the 28th: 28 − 21 − 1 = 6.
A secret agent wants to crack a six-digit code. He knows that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions. Which of the following numbers is the code? (Each ? stands for an unknown digit.)
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Answer: D — 12?9?8
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Hint 1 of 3
For each number, circle the digits in the 1st, 3rd and 5th spots, and box the digits in the 2nd, 4th and 6th spots.
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Hint 2 of 3
The known digits in one group might already be too big for the other group to ever catch up, even using a 9 in each blank.
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Hint 3 of 3
Look for the one number whose two groups CAN be made equal.
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Approach: compare the known digits in the two groups and see which one can balance
Add up the known digits in the odd spots (1st, 3rd, 5th) and in the even spots (2nd, 4th, 6th), and remember a blank can be at most 9.
In A, B, C and E one group's known digits are already so far ahead that even filling the other group's blanks with 9 cannot make them equal.
In D the number is 12?9?8: the even spots give 2 + 9 + 8 = 19, and the odd spots are 1 + ? + ?, which reaches 19 when both blanks are 9 (1 + 9 + 9 = 19).
Only option D can have its two groups equal, so the code is option D.
Meta collects pictures of famous sports people. Each year she collects as many pictures as she did in the previous two years. In 2008 she had 60 photos and this year she has 96. How many photos did she have in 2006?
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Answer: B — 24
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Hint 1 of 2
Each year's count equals the previous two years added together.
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Hint 2 of 2
You know 2008 and 2009; work backwards to find 2007, then 2006.
In a vase there is one red, one blue, one yellow and one white flower. Maja the bee visits each flower exactly once. She begins with the red flower and she never flies directly from the yellow to the white flower. In how many different ways can she visit each flower?
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Answer: D — 4
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Hint 1 of 2
She always starts at the red flower, so list the orders of the other three.
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Hint 2 of 2
Then cross out any order where yellow comes immediately before white.
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Approach: list the routes and remove the forbidden ones
Starting at red, the other three flowers (blue, yellow, white) can be ordered in 6 ways.
Remove every order in which yellow is immediately followed by white.
Two of the six orders are forbidden, leaving 4 allowed routes.
In a haunted house the house ghost suddenly disappears. At that moment in time all clocks show 6:15. However, there is also one strange clock in the house that showed the correct time before that event — starting from the disappearance it begins to count backwards. At 19:30 in real time the house ghost reappears. What time does the odd clock show at that moment?
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Answer: A — 17:00
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Hint 1 of 2
The vanishing happens at 6:15 in the evening, that is 18:15.
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Hint 2 of 2
Find how much real time passes, then subtract it because the odd clock runs backwards.
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Approach: run the time backwards by the elapsed amount
When the ghost vanishes the clocks read 6:15 p.m. = 18:15.
The ghost returns at 19:30, so 1 hour 15 minutes of real time pass.
The odd clock counts backwards, so it shows 18:15 − 1:15 = 17:00.
Sylvia draws shapes made of straight lines that are each 1 cm long. At the end of each line she turns a right angle, either left or right. At every turn she writes down a ♥ or a ♠, and the same symbol always means a turn in the same direction. Today her notes show ♥♠♠♠♥♥. Which of the following shapes could she have drawn today if A is her starting point?
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Answer: E
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Hint 1 of 3
Every symbol is a right-angle turn; one symbol always turns the same way and the other symbol always turns the other way.
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Hint 2 of 3
Notice the notes have three of the same symbol in a row (♠♠♠) in the middle, so the correct shape must make three same-direction turns in a row there.
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Hint 3 of 3
Start at A, walk 1 cm at a time, and turn the way each symbol tells you.
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Approach: match the ♥/♠ turn pattern by walking from the start point A
Each line is 1 cm and each symbol is a right-angle turn, with one symbol always turning left and the other always turning right.
The middle of the notes has three of the same symbol in a row (♠♠♠), which means three turns the same way in a row — that traces three sides of a little square.
Starting at A and walking the path while turning as ♥♠♠♠♥♥ tells you, the path closes up in the shape of option E.
In Funny-Foot-Land men and women wear the same sort of shoes. Each man has a left foot that is two sizes bigger than his right foot. Each woman has a left foot that is one size bigger than her right foot. However, shoes are only sold in pairs of the same size. To save money some friends decide to buy shoes together. After putting on their new shoes, two shoes are left over — one of size 36 and one of size 45. What is the minimum number of people in that group?
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Answer: A — 5
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Hint 1 of 2
Each person uses two different shoe sizes; shoes come only in same-size pairs.
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Hint 2 of 2
Think of going from size 36 up to size 45 in steps of 1 (a woman) or 2 (a man) — how few steps reach 45?
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Approach: link size 36 to size 45 with the fewest people
A man uses sizes that differ by 2; a woman uses sizes that differ by 1.
Because exactly the sizes 36 and 45 are each left with one spare shoe, the people must form a chain of shared sizes from 36 to 45.
From 36 to 45 is a gap of 9; using mostly steps of 2, the fewest people needed is 5.
