213, 214 and 215 are three numbers in a row, each one more than the one before. Mohammad writes three numbers like that, but with four digits each. His sister erases some digits from each number, leaving:
???7, ?898, 48??
Which digits (from left to right) are missing?
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Answer: D — 4 8 9, 4, 9 9
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Hint 1 of 2
The three numbers are consecutive, and the middle one looks like _898.
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Hint 2 of 2
Pick the four-digit number ending 898, then write the one before and the one after it.
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Approach: identify three consecutive numbers from the visible digits
The middle number is 4898 (form _898, and it sits between numbers ending 7 and starting 48).
If you add up the digits of the year 2016 (2 + 0 + 1 + 6), the result is 9. What is the next year after 2016 for which the sum of the digits is 9 again?
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Answer: B — 2025
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Hint 1 of 2
The next year must be after 2016 and have digits adding to 9.
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Hint 2 of 2
Try years just after 2016 and add their digits.
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Approach: check years after 2016 for digit sum 9
Add the digits of each year right after 2016: 2017 gives 10, 2018 gives 11, and they keep climbing, so none of 2017–2024 lands back on 9.
Marie wants to put the digit 3 somewhere into the number 2014. Where must she put the 3 so that the new number (with all 5 digits) is as small as possible?
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Answer: D — between 1 and 4
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Hint 1 of 3
A number is smaller when its first (left-most) digits are smaller.
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Hint 2 of 3
Putting the big digit 3 near the front pushes the number up, so push the 3 as far right as you can.
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Hint 3 of 3
Write out each new number and read them like words to see which comes first.
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Approach: keep the small left-hand digits and push the 3 to the right
Try the 3 in each gap: 32014, 23014, 20314, 20134, 20143.
Reading them like a race, the one that stays smallest the longest at the front is 20134.
Five children are talking about the number 325. Andreas: “It is a three-digit number.” Boris: “All the digits are different.” Sara: “The digit sum is 10.” Gerda: “The units digit is 5.” Daniela: “All the digits are odd.” Who has made a mistake?
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Answer: E — Daniela
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Hint 1 of 2
Check each child's statement against the actual number 325.
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Hint 2 of 2
One statement is simply false — look at whether every digit is really odd.
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Approach: test each claim about 325
325 is a three-digit number (Andreas correct), its digits 3, 2, 5 are all different (Boris correct).
3+2+5 = 10 (Sara correct) and the units digit is 5 (Gerda correct).
But 2 is even, so 'all the digits are odd' is wrong — that is Daniela.
Mia has 3 cards, each showing a three-digit number. When she adds the three numbers she gets 782. Sadly a worm has eaten one digit on each card, so they now read 2 ? 3, 1 ? 4 and 4 1 ?. What do you get when you add the three digits the worm ate?
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Answer: D — 11
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Hint 1 of 3
First add up only the digits you can still see, putting a 0 in each eaten spot.
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Hint 2 of 3
Compare that total with 782 to see how much the eaten digits must add back.
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Hint 3 of 3
Remember: an eaten digit in a tens place is worth that many tens, and an eaten digit in a ones place is worth that many ones.
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Approach: add the visible digits first, then see how much the eaten digits must add back
Treat each eaten spot as 0: the cards read 203, 104 and 410, which add to 717.
But the real total is 782, so the eaten digits must add back 782 − 717 = 65.
Two of the eaten digits sit in tens places, so together they are worth 60 (meaning those two digits add to 6); the third sits in a ones place worth 5 (so that digit is 5).
The three eaten digits therefore add to 6 + 5 = 11 (D).
7 cards are arranged as shown. Each card has 2 numbers on it, with 1 of them written upside down. The teacher wants to rearrange the cards so that the sum of the numbers in the top row is the same as the sum of the numbers in the bottom row. She can do this by turning one of the cards upside down. Which card must she turn?
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Answer: E — G
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Hint 1 of 2
Add up the top row and add up the bottom row, then see how far apart the two totals are.
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Hint 2 of 2
Turning one card swaps its top and bottom numbers, so look for the card whose swap moves exactly half the gap from one row to the other.
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Approach: compare the two row totals and flip the one card that evens them out
The top numbers add to 7+5+4+2+8+3+2 = 31 and the bottom numbers add to 4+3+5+5+7+7+4 = 35, a gap of 4 (the bottom is 4 bigger).
All 14 numbers add to 66, so to make the rows equal each must be 66 / 2 = 33; the top needs to gain 2 and the bottom to lose 2.
Flipping a card moves its top number down and its bottom number up, so we need a card whose bottom is 2 more than its top — that is card G (top 2, bottom 4).
A box has fewer than 50 cookies in it. The cookies can be divided evenly between 2, 3, or 4 children. However, they cannot be divided evenly between 7 children, because 6 more cookies would be needed. How many cookies are there in the box?
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Answer: D — 36
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Hint 1 of 2
Sharing evenly between 2, 3 and 4 children means the number is in the 2, 3 and 4 times tables — so it is in the 12 times table.
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Hint 2 of 2
List the multiples of 12 under 50, then check which one becomes a multiple of 7 after you add the 6 missing cookies.
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Approach: list multiples of 12 under 50, then test the sharing-by-7 clue
To share evenly between 2, 3 and 4 children the number must be in all three times tables, which means the 12 times table: 12, 24, 36, 48.
Needing 6 more cookies to share between 7 means that number plus 6 lands in the 7 times table.
Check each: 12+6=18, 24+6=30, 36+6=42, 48+6=54 — only 42 is in the 7 times table (6 x 7), so the number is 36.
The pages of a book are numbered with 1, 2, 3, 4, 5 and so on. The digit 5 appears exactly 16 times. What is the maximum number of pages the book can have?
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Answer: B — 64
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Hint 1 of 2
Count where the digit 5 shows up as you list page numbers 1, 2, 3, ...
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Hint 2 of 2
The block 50-59 alone contributes ten 5s (the tens digit); add those to the single 5s like 5, 15, 25, ...
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Approach: count occurrences of the digit 5
Up to 49 the 5s appear at 5, 15, 25, 35, 45: five of them.
The block 50-59 has a 5 in every tens digit: ten more, plus the extra units-5 in 55, reaching the 16th by page 59.
Pages 60-64 add no new 5s, but page 65 would add a 17th.
So the most pages with exactly sixteen 5s is 64 (B).
Leo has built a stick made up of 27 building blocks (see picture). He splits the stick into two pieces so that one part is twice as long as the other. He keeps repeating this: each time he takes one of the two pieces and splits it so that one piece is twice as long as the other. Which of the following pieces can never result in this way? (The choices are pieces of length 2, 4, 6, 8 and 10 blocks.)
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Answer: E — 10
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Hint 1 of 2
Splitting a piece into a 2:1 ratio only works when its length divides into three equal parts.
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Hint 2 of 2
List every length you can reach starting from 27 and see which option never appears.
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Approach: track which lengths a 2:1 split can produce
27 splits into 18 and 9; 18 splits into 12 and 6; 9 splits into 6 and 3; 12 splits into 8 and 4; 6 splits into 4 and 2.
The reachable lengths are 2, 3, 4, 6, 8, 9, 12, 18, 27.
A 10-block piece never appears, so 10 can never result.
In each box exactly one of the digits 0, 1, 2, 3, 4, 5 and 6 is to be written. Each digit is used only once. The picture on the right shows two 2-digit numbers being added to give a 3-digit number. Which digit has to be written in the grey box so that the sum is correct?
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Answer: D — 5
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Hint 1 of 3
The answer has 3 digits, so the two 2-digit numbers must add up to at least 100.
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Hint 2 of 3
You only have the digits 0 to 6 once each, so the hundreds digit of the answer can only be 1.
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Hint 3 of 3
Once you know the answer starts with 1, fit the remaining digits and read the grey (ones) box.
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Approach: find the only valid sum, then read the grey (units) box
Two 2-digit numbers add to a 3-digit number using 0..6 once each.
The only working sum is 105 (e.g. 42 + 63), using digits 0,1,2,3,4,5,6.
The grey box is the units digit of the result, which is 5.
The number 35 has a special property: it can be divided exactly by its units digit, because \(35 \div 5 = 7\). The number 38 does not have this property. How many numbers bigger than 21 but smaller than 30 have this property?
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Answer: B — 3
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Hint 1 of 3
Bigger than 21 and smaller than 30 means the numbers 22, 23, 24, 25, 26, 27, 28, 29.
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Hint 2 of 3
For each one, can you split it into equal groups of its last digit with none left over?
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Hint 3 of 3
Count how many of the eight numbers pass that test.
Gregory made two 3-digit numbers from the digits 1, 2, 3, 4, 5, 6. Each digit was used only once. Afterwards he added the two numbers together. What is the largest answer he could have got?
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Answer: D — 1173
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Hint 1 of 2
To make a sum large, put the biggest digits where they count the most.
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Hint 2 of 2
Give the two hundreds places the largest digits, then the tens, then the units.
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Approach: place the largest digits in the highest places
Use 6 and 5 in the hundreds places, 4 and 3 in the tens, 2 and 1 in the units.
The two numbers add to (600 + 500) + (40 + 30) + (2 + 1).
The teacher said, “In our school library there are roughly 2010 books.” The pupils then guessed exactly how many there are. Artur guessed 2010, Beate guessed 1998 and Carlos guessed 2015. Their guesses are off by 12, 7 and 5, but not in that order. How many books are in the library?
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Answer: A — 2003
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Hint 1 of 2
The real number differs from the three guesses by 12, 7 and 5 in some order.
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Hint 2 of 2
Try a value near 2010 and check that its distances to 2010, 1998 and 2015 are exactly 12, 7 and 5.
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Approach: find the value whose distances to the guesses are 12, 7, 5
Berti’s friends each add together the day and the month of their birthday. They all get the answer 35, but no two of them have the same birthday. What is the largest number of friends Berti can have?
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Answer: B — 8
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Hint 1 of 2
You need months and days with month + day = 35, and each birthday must be a real date.
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Hint 2 of 2
Start from December and step down through the months, checking the day fits that month.
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Approach: count valid (month, day) pairs summing to 35
List month + day = 35 with a valid day: Dec 23, Nov 24, Oct 25, Sep 26, Aug 27, Jul 28, Jun 29, May 30.
April would need day 31, which doesn't exist, and earlier months need impossible days.
That gives 8 different birthdays, so at most 8 friends.
13 children registered for a competition. Then another 19 joined. Six equally big teams are needed for the competition. How many more children are needed, so that six equally big teams can be formed?
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Answer: D — 4
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Hint 1 of 2
First add the two groups of children together.
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Hint 2 of 2
Six equal teams means the total must split into 6 equal piles, so count up by sixes past your total.
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Approach: count up by sixes to the first number past the total
Altogether there are 13 + 19 = 32 children.
Six equal teams need a total that shares evenly into 6 piles, so count by sixes: 6, 12, 18, 24, 30, 36.
Johannes has only 5 Cent coins and 10 Cent coins in his pocket. Altogether he has 13 coins. Which of the following amounts cannot be the total of his coins?
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Answer: B — 60 c
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Hint 1 of 2
With 13 coins, swapping a 5 c coin for a 10 c coin changes the total by a fixed step.
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Hint 2 of 2
Find the smallest and largest possible totals, and the step between reachable totals.
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Approach: range and step of the total
All 5 c gives 65 c; all 10 c gives 130 c; each 5→10 swap adds 5 c, so totals are 65, 70, 75, …, 130.
80, 70, 115 and 125 are all multiples of 5 in that range, but 60 c is below 65 c, so it cannot be the total, answer B.
Jonas and Elias went to the beach for their vacation, where they had ice cream every day. Each ice cream they had, had two or three balls. On the last day of vacation, Jonas and Elias had had 23 and 19 ice cream balls in total, respectively. At least how many days were they on vacation?
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Answer: C — 8
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Hint 1 of 3
To use up the fewest days, give the bigger eater (Jonas, 23 balls) the most balls each day.
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Hint 2 of 3
Even eating 3 balls every single day, count how many days it takes to reach 23.
Still stuck? Show hint 3 →
Hint 3 of 3
The same number of days has to work for both boys, so check that 19 also fits in that many days.
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Approach: give the most balls per day to use the fewest days, then check both totals fit
To finish in as few days as possible, eat the biggest ice cream (3 balls) every day.
Counting by 3 toward Jonas's 23: seven days give only 21 balls, which is not enough, so they need at least 8 days.
Eight days really works for both: Jonas eats 3 balls on 7 days and 2 balls on 1 day (21 + 2 = 23), and Elias eats 3 balls on 3 days and 2 balls on 5 days (9 + 10 = 19).
So they were on vacation at least 8 days, choice C.
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.
Still stuck? Show hint 2 →
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.
Still stuck? Show hint 3 →
Hint 3 of 3
Look for the one number whose two groups CAN be made equal.
Show solution
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.
