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Sometimes a number likes to hide. We draw a little box ? where it sits.
Your job is to find what the box is worth. You are a number detective!
Detectives have tricks. We will spot the jump in a pattern. We will balance two sides like a see-saw. We will turn a story into numbers. We will undo steps to walk backwards. And when nothing pops out, we will try every clue in order. No tricky letters — small numbers and pictures.
What comes next?
Some numbers march in a line. Watch them go.
2, 4, 6, then a box. Can you feel where it wants to go?
Each hop is the same: +2. So the box is 6 + 2 = 8.
The secret is the hop. Ask: how do I get from one number to the next? Find that, and you can keep going forever.
Let's try one that hops by a bigger amount:
3 to 7 is +4. 7 to 11 is +4. So the box is 11 + 4 = 15.
THE HOP. Find how far it jumps between two numbers. If every jump is the same, keep jumping the same way.
Do not stare at one number and guess. Look at TWO numbers next to each other to find the hop. One number alone never tells you the rule.
Eli drew a board on the floor with nine squares and wrote a number in each one, starting from 1 and adding 3 each time, until the board was full. Three of the numbers she wrote are shown in the picture. Which number below could be the one in the colored box?
Eli starts at 1. She adds 3 each time. So the hop is +3.
Now skip-count and write each number down: 1, 4, 7, 10, 13, 16, 19, 22, 25.
Look at the choices. Which one is in her list? 14 is not. 17 is not. 20 is not. But 22 is! So the answer is 22.
I do not need to be clever. I keep adding 3 and write each number down. Then I look for the choice that matches.
Find the hop. Then keep hopping.
2009 · #9 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...
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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- 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.
- To erase the gap of 6, that takes 6 weeks.
- After 6 weeks the numbers are equal (37 each).
2010 · #15 Hans started a chain e-mail. He sent an e-mail to his friend Peter, who sent it on to 2 more people. Each person who gets the e-mail...
Hans started a chain e-mail. He sent an e-mail to his friend Peter, who sent it on to 2 more people. Each person who gets the e-mail sends it on to 2 more people. After 3 rounds, 1 + 2 + 4 = 7 people have received the e-mail. How many people have received the e-mail after 5 rounds?
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- Each round the number of new people doubles: 1, 2, 4, then 8, then 16.
- Add up all five rounds: 1 + 2 + 4 + 8 + 16 = 31.
- So 31 people have the e-mail, choice C.
Make both sides match
A see-saw is fair when both sides weigh the same. It sits flat.
The left side is 3 + 4 = 7. So the right side must also be 7.
The right side is ? + 2. So ? + 2 = 7. The box is 5, because 5 + 2 = 7. To find the box, take away the part you know: 7 take away 2 is 5.
An even faster way: cover the twins
Sometimes the same thing sits on BOTH sides. You can hide those twins with your finger. Whatever is left over must still match.
Two triangles and a 6 on the left. Four triangles on the right. Cover two triangles on each side. Now 6 is left on one side, and two triangles on the other. So two triangles make 6, which means one triangle is 3.
Both sides must be equal. Add up the side you know. Then make the other side the same. If the same shape sits on both sides, cover the twins first.
Do not copy a number straight across. Add up the whole side first, then fill the box.
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?
First clue: red triangle + 4 = 7. Take away 4. So the triangle is 3.
Second clue: blue square + triangle = 9. The triangle is 3, so square + 3 = 9.
Take away 3. The square is 6.
I solve the easy clue first. Once I know the triangle, I drop it into the next line and the square pops right out.
Find the easy shape first. Then use it to find the next one.
2010 · #9 Suppose ▲ + ▲ + 6 = ▲ + ▲ + ▲ + ▲. Which number should replace ▲?
Suppose ▲ + ▲ + 6 = ▲ + ▲ + ▲ + ▲. Which number should replace ▲?
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- The left side is two triangles and a 6; the right side is two triangles and two more triangles.
- Cover the two matching triangles on each side, and the 6 is left on the left while two triangles are left on the right.
- So two triangles make 6, which means one triangle is 6 split into 2 equal parts, or 3.
- The triangle is 3, choice B.
2017 · #10 Four apples and one pear weigh as much as three pears. What is therefore correct?
Four apples and one pear weigh as much as three pears. What is therefore correct?
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- 4 apples + 1 pear balances 3 pears.
- Remove one pear from each side: 4 apples balance 2 pears.
- Halve both sides: 2 apples balance 1 pear.
- So two apples weigh as much as one pear: E.
Quick check: patterns and fair sides
Find the hop, then keep both sides equal.
2022 · #4 Which two numbers can replace the two boxes in 2022 + □ = 2020 + □ to make it true?
Which two numbers can replace the two boxes in 2022 + □ = 2020 + □ to make it true?
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- Since 2022 is 2 more than 2020, the right box must hold 2 more than the left box.
- Among the pairs, 3 and 5 differ by 2 (add 3 on the left, 5 on the right).
- Check: 2022 + 3 = 2025 and 2020 + 5 = 2025.
- So the answer is 3 and 5.
2010 · #9 Suppose ▲ + ▲ + 6 = ▲ + ▲ + ▲ + ▲. Which number should replace ▲?
Suppose ▲ + ▲ + 6 = ▲ + ▲ + ▲ + ▲. Which number should replace ▲?
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- The left side is two triangles and a 6; the right side is two triangles and two more triangles.
- Cover the two matching triangles on each side, and the 6 is left on the left while two triangles are left on the right.
- So two triangles make 6, which means one triangle is 6 split into 2 equal parts, or 3.
- The triangle is 3, choice B.
Turn a story into a number sentence
A word problem is a little story. We turn the story into a number sentence with a box.
Say you have 6 balloons. You give some away. You have 2 left.
The sentence is 6 − ? = 2. What did you give away? You gave 4, because 6 − 4 = 2.
Read the story slowly. Drop the numbers in. Put a box where the missing number hides. Some words tell you what to do:
Words become a number sentence. Put a box where the missing number hides. Then fill the box.
Watch the words. More means add. Gave away or left means take away. Read it twice before you write.
A sandwich and a juice cost 12 Euros together. A sandwich and two juices cost 14 Euros together. How many Euros does one juice cost?
A sandwich and a juice cost 12. A sandwich and TWO juices cost 14.
Both have one sandwich. The only change is one extra juice.
The cost went up by 14 − 12 = 2. So one juice costs 2 Euros.
I line up the two facts. They are almost twins. I look at what is different, and that gives me the juice.
Compare two facts. The bit that changed tells you the missing piece.
2012 · #10 3 balloons cost 12 cents more than 1 balloon. How much does 1 balloon cost?
3 balloons cost 12 cents more than 1 balloon. How much does 1 balloon cost?
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- 3 balloons cost 12 cents more than 1 balloon, so the extra 2 balloons cost 12 cents.
- Then 1 balloon costs 12 / 2 = 6 cents.
- So one balloon costs 6 cents.
2009 · #17 A white and a black pig weigh together 320 kg. The black pig weighs 32 kg more than the white one. How much does the white pig weigh?
A white and a black pig weigh together 320 kg. The black pig weighs 32 kg more than the white one. How much does the white pig weigh?
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- Together the pigs weigh 320 kg, so equal pigs would be 160 kg each.
- The black pig is 32 kg more than the white pig.
- Give half of 32 (which is 16) to the black and take 16 from the white: white = 160 − 16 = 144 kg.
- The white pig weighs 144 kg.
When stuck, try every clue
Sometimes a story is tricky and no box pops out fast. That is okay! You still have a power move.
Try every choice, one at a time, and check each one. Go in order so you do not miss one.
Say the answer is between 1 and 5, and it has to make 7 work. Walk down the list:
You did not need a clever trick. You checked them in order until one fit. With multiple-choice, the answer is right there on the page — so you only have five things to test.
Test the choices in order. Try the first one. Check it. If it fails, try the next. Stop when one fits.
Do not skip around or guess wildly. Go down the list in order and check each one, or you might miss the right one.
Every night the wizard Tilim makes the weather forecast for the king. When Tilim gets it right he gets 3 gold coins, but when he makes a mistake he pays a fine of 2 gold coins. After making the prediction for 5 days, Tilim found that he neither won nor lost coins. How many times did he get the weather forecast right in those 5 days?
The wizard guesses the weather for 5 nights. A right night wins 3 coins. A wrong night pays 2 coins. In the end his coins did not change.
So the coins he won must match the coins he paid. Let's test how many right nights, in order.
Try right = 1: he wins 1 × 3 = 3 coins, but loses 4 × 2 = 8 coins. He lost coins. No.
Try right = 2: he wins 2 × 3 = 6 coins, and loses 3 × 2 = 6 coins. 6 and 6 match! Nothing changed. Yes!
So he was right 2 nights.
I do not try to be fancy. I test 1, then 2, and check if the coins balance. The second try snaps right into place.
No quick idea? Try the choices in order and check each. The list is short.
2014 · #15 Hubert the rabbit loves cabbages and carrots. In one day he eats either 9 carrots, or 2 cabbages, or one cabbage and 4 carrots. In one...
Hubert the rabbit loves cabbages and carrots. In one day he eats either 9 carrots, or 2 cabbages, or one cabbage and 4 carrots. In one week Hubert ate 30 carrots. How many cabbages did he eat during this week?
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- Only two kinds of day give carrots: a 9-carrot day, or a day of 1 cabbage and 4 carrots.
- Two 9-carrot days give 18 carrots, and three of the ‘1 cabbage + 4 carrots’ days give 12 more — that is 18 + 12 = 30 carrots, using 5 days.
- Those three mixed days give 3 cabbages, and the 2 days left over are 2-cabbage days, giving 2 × 2 = 4 more.
- Total cabbages = 3 + 4 = 7.
2023 · #13 In a queue in front of a ferry there are 8 cars with 19 people in total. There are either 2 or 3 people in each car. How many cars are...
In a queue in front of a ferry there are 8 cars with 19 people in total. There are either 2 or 3 people in each car. How many cars are there with exactly 2 people?
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- Eight cars with 2 people each would carry 16 people, but there are 19.
- The 3 extra people come from 3 cars holding 3 instead of 2.
- So 8 − 3 = 5 cars hold exactly 2 people.
Work backwards to find the start
Most puzzles go forwards. But sometimes you know the END and you want the START.
So you walk backwards and undo each step.
The step was add 4, and you ended at 10. To undo an add, you take away. 10 − 4 = 6. The start was 6.
Undo is like rewinding. Watch the pairs of opposites:
Undo means flip it. Add flips to take away. Take away flips to add.
For big kids: a longer rewind
A number was doubled, then 1 was added, and it landed on 11. Walk backwards: undo the +1 first (11 − 1 = 10), then undo the doubling (10 split into 2 is 5). The start was 5. Always undo the LAST step first.
Start at the end and undo every step. Flip each step the other way, and undo the last step first.
Going backwards, you must flip the step. If they added, you take away. Do the same step again and you will go the wrong way.
A dog has 2 puppies that both weigh the same. Picture 1 shows that the dog and one puppy together weigh 14 kilograms. Picture 2 shows that the dog and both puppies together weigh 18 kilograms. How many kilograms does the big dog weigh?
Picture 1: the dog and one puppy weigh 14 kg. Picture 2: the dog and BOTH puppies weigh 18 kg.
The two pictures are twins, except one has an extra puppy. Going from 14 to 18 added one puppy. So one puppy weighs 18 − 14 = 4 kg.
Now undo. Take that puppy back off Picture 1: 14 − 4 = 10 kg. The big dog weighs 10 kg.
The extra puppy is the only difference, so it tells me one puppy's weight. Then I peel a puppy back off to leave the dog alone.
Spot the one extra thing. That gives you its weight. Then take it away to find the rest.
2010 · #11 Matthias and Klara live in a tower block. Klara lives 12 floors above Matthias. One day Matthias climbs the stairs to visit Klara. When...
Matthias and Klara live in a tower block. Klara lives 12 floors above Matthias. One day Matthias climbs the stairs to visit Klara. When he is halfway there he is on the 8th floor. On which floor does Klara live?
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- Half of the 12-floor climb is 6 floors, and that point is the 8th floor.
- So Matthias starts on the 8 − 6 = 2nd floor.
- Klara lives 12 floors higher: 2 + 12 = 14th floor.
2017 · #16 Which number must be written into the circle with the question mark so that the calculation is correct?
Which number must be written into the circle with the question mark so that the calculation is correct?
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- Following the arrows must return to the starting circle.
- One step is 'x0', whose result is 0, so the circle it leads to holds 0.
- From there: 0, then +6 gives 6, then x4 gives 24, then -15 gives the question mark.
- So the question mark is 24 - 15 = 9.
Quick check: sentences, trying, and undoing
Turn words into a box, test the choices, then work backwards.
2012 · #10 3 balloons cost 12 cents more than 1 balloon. How much does 1 balloon cost?
3 balloons cost 12 cents more than 1 balloon. How much does 1 balloon cost?
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- 3 balloons cost 12 cents more than 1 balloon, so the extra 2 balloons cost 12 cents.
- Then 1 balloon costs 12 / 2 = 6 cents.
- So one balloon costs 6 cents.
2025 · #8 A dog has 2 puppies that both weigh the same. Picture 1 shows that the dog and one puppy together weigh 14 kilograms. Picture 2 shows...
A dog has 2 puppies that both weigh the same. Picture 1 shows that the dog and one puppy together weigh 14 kilograms. Picture 2 shows that the dog and both puppies together weigh 18 kilograms. How many kilograms does the big dog weigh?
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- Picture 1 is the dog and one puppy (14 kg). Picture 2 is the dog and both puppies (18 kg).
- Going from 14 kg to 18 kg adds one more puppy, so one puppy weighs 4 kg.
- Take that puppy away from picture 1: 14 take away 4 leaves 10 kg for the dog. The answer is B.
Growing picture patterns
Some pictures grow. Each new picture adds the same little bit.
Watch how many dots are added each time.
The dots go 1, 2, 3, then 4. Each picture adds 1 more dot. So the next picture has 4 dots. This is the same HOP idea from Chapter 1 — now it is hiding inside a picture.
One gotcha: sometimes neighbors share. Watch the towels in the worked problem — each new towel adds only one peg, because it shares a peg with the towel next to it.
Count how many are added each time. Then add that same amount once more. Do not recount the whole picture.
Do not count from scratch every time. Find the step the picture grows by — and watch out, sometimes neighbors share a piece.
Father hangs towels on the washing line as shown in the picture. For three towels he uses 4 clothes pegs. How many clothes pegs would he use for 5 towels?
3 towels use 4 pegs. There is one extra peg because the end needs a peg too.
Each new towel shares a peg with its neighbor. So each new towel adds only 1 peg.
Go from 3 towels to 5 towels: that is 2 more towels, so 2 more pegs. 4 + 2 = 6 pegs.
I see that pegs grow by one each time a towel joins. So I do not recount — I add one peg per new towel.
Find the step, then add steps. Here, pegs are always one more than towels.
2019 · #9 Linda fixes 3 photos on a pin board next to each other. She uses 8 pins to do so (see picture). Peter wants to fix 7 photos in the same...
Linda fixes 3 photos on a pin board next to each other. She uses 8 pins to do so (see picture). Peter wants to fix 7 photos in the same way. How many pins does he need for that?
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- The top row of pins has one more pin than the number of photos, and so does the bottom row.
- For 3 photos that is 4 pins on top and 4 pins on the bottom, which makes 8 pins — this matches the picture.
- For 7 photos it is 8 pins on top and 8 pins on the bottom.
- Counting both rows gives 8 + 8 = 16 (B).
2011 · #24 Sylvia draws patterns with hexagons as shown. If she carries on drawing in this way, how many hexagons will there be in the fifth pattern?
Sylvia draws patterns with hexagons as shown. If she carries on drawing in this way, how many hexagons will there be in the fifth pattern?
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- Count the pictures: pattern 1 has 1 hexagon, pattern 2 has 7, pattern 3 has 19.
- Each new ring adds 6 more than the last: +6 to get 7, +12 to get 19, so next is +18, then +24.
- Pattern 4 = 19 + 18 = 37, and pattern 5 = 37 + 24 = 61 hexagons, answer D.
Number Detective Test
Six missing-number puzzles, easy first. Take your time.
2022 · #4 Which two numbers can replace the two boxes in 2022 + □ = 2020 + □ to make it true?
Which two numbers can replace the two boxes in 2022 + □ = 2020 + □ to make it true?
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- Since 2022 is 2 more than 2020, the right box must hold 2 more than the left box.
- Among the pairs, 3 and 5 differ by 2 (add 3 on the left, 5 on the right).
- Check: 2022 + 3 = 2025 and 2020 + 5 = 2025.
- So the answer is 3 and 5.
2015 · #5 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....
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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- 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).
- The hidden number is 6.
2015 · #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...
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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- 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.
- The answer is 3.
2009 · #9 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...
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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- 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.
- To erase the gap of 6, that takes 6 weeks.
- After 6 weeks the numbers are equal (37 each).
2009 · #17 A white and a black pig weigh together 320 kg. The black pig weighs 32 kg more than the white one. How much does the white pig weigh?
A white and a black pig weigh together 320 kg. The black pig weighs 32 kg more than the white one. How much does the white pig weigh?
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- Together the pigs weigh 320 kg, so equal pigs would be 160 kg each.
- The black pig is 32 kg more than the white pig.
- Give half of 32 (which is 16) to the black and take 16 from the white: white = 160 − 16 = 144 kg.
- The white pig weighs 144 kg.
2019 · #12 All dogs are equally heavy. The two balance scales are shown in the picture. How much could one dog weigh?
All dogs are equally heavy. The two balance scales are shown in the picture. How much could one dog weigh?
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- The first scale tips toward the 12 kg weight, so one dog must weigh a little less than 12 kg.
- The second scale tips toward the two dogs, so the two dogs together weigh more than 20 kg, which means one dog weighs more than 10 kg.
- The only whole number of kilograms that is more than 10 but less than 12 is 11 kg (E).
Quick reference
- For patterns, find the hop first, then keep hopping.
- Both sides of an = sign must match. Cover the twins.
- To undo add, you take away. To undo take away, you add.
- Stuck? Try the choices in order and check each one.