The following problems are taken from different Math Contests and Olympiads as well as on-line open sources.
Most of them were very challenging problems during Math Chicago Classes and were explained in depth.
Problem 1
1 2 3 4 5 6 7 8 9 = 100
The numbers above are part of an equation, in which all the plus and minus signs from the left part of the equation have been removed. What's more, two of the digits are actually part of a two digit number. Can you put the pluses and minuses into to make the equation work?
Problem 2
What digits would you plug into the letters so the addition would make sense?
A B O U T
+ R A I N
T H I N K
Problem 3
In how many ways can 4 people be seated at a round table?
Problem 4
John has 9 balls in his bag. At least one of them is black. From every 4 balls at least two are of the same color, and from every 5 balls at most three are of the same color. How many black balls are in the bag?
Problem 5
Can you find the logic of this puzzle and figure out which character comes next?
Problem 6
How many solutions can you find for the subtraction below?
K A N G A
– R O O
R I N G
Problem 7
Can you arrange the order of the seven strips so that each row contains a correct mathematical statement? The strips that have the operators can be flipped if needed.
Problem 8
You embark on a voyage to a mythical land. All of the inhabitants there are either ‘knights’ or ‘knaves’... ‘Knights’ always tell the truth, whereas ‘knaves’ always lie. You walk up to two inhabitants, Adam and Bert and ask “who are you”? . . . Adam replies “we are both knaves”. What are they in fact?
Problem 9
What number is hidden under the question mark below?
Problem 10
Five people live in 5 houses. The houses are in a line next to each other. Every house has its own color. The owners drive 5 different cars and like 5 different drinks. They have 5 different kinds of animals.
The Norwegian lives in the first house.
The middle house owner drinks milk.
The Norwegian lives next to the blue house.
The green house owner drinks coffee.
The green house is left to the white house.
The Englishman lives in the red house.
The German drinks tea.
The yellow house owner drives Dodge.
The Toyota driver lives next to the cat owner.
The Dane drive Ford
The neighbor of the Toyota driver drinks water.
The Jeep owner has a bird.
The Swede has a dog.
The horse owner lives next to the yellow house.
The Audi driver drinks beer.
QUESTION: Who has the fish? There is only one possible answer.
Problem 11
A man has to take a hen, a fox, and some corn across a river. He can only take one thing across at a time. Unless the man is present the fox will eat the hen and the hen eat the corn. How is it done?
Problem 12
Alan, Bill and Chris dug up 9 nuggets. Their weights were 154, 16, 19, 101, 10, 17, 13, 46 and 22 kg. They took 3 each. Alan's weighed twice as much as Bill's. How heavy were Chris's nuggets?
Problem 13
The product of 3 brothers' ages is 175. Two are twins. How
old is the other one?
Problem 14
The sum uses all the digits from 0 to 9 and the subtraction uses all the digits from 1 to 9.
2 8 *
+ * * 4
* * * *
9 * *
– * 4 *
* * 1
Problem 15
Express each of the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 using exactly four digits of 4 and +, –, x, ÷ and ( ).
Problem 16
At a party, a guest asks the age of the hosts 3 children. The host tells the guest the following information:
Assume each kid is a counting number age;
The product of their ages is 72;
The sum of their ages is the same as my house number.
The guest goes outside and
looks at the house number. Upon returning, the guest asks for additional
information. The host then says "Oops, I forgot to tell you that the oldest
likes strawberry ice cream."
You now have all the information to determine the ages of the kids.
How old are they, and why?
Problem 17
How many squares are on an 8 x 8 checker board?
Problem 18
Here is a form of an old puzzle that was first seen in Ancient Rome. In the picture the stick dog is facing to the left. By moving only two of the sticks can you make him face right?
Problem 19
What number should logically replace the question mark in the box?
Problem 20
The letters around the four circles are arranged in a logical pattern. Can you determine the pattern and figure out which letters should replace the question marks on the first circle?
Problem 21
Can you determine the pattern and figure out what two letters would follow up?
O T T F F S S E _ _
Problem 22
A very hungry Math Kangaroo 2007 Winner having 10 bucks in his pocket walks into 7-11 store. He purchases 7-Eleven Sandwich, 7-Eleven Bakery Muffin, 7-Eleven Frozen Dessert and a bottle of water. The clerk at the counter multiplies the four prices and informs the winner that the total cost of it is $7.11. He was completely surprised that the cost was the same as the name of the store. Smart Math Kangaroo Winner calmly informs the clerk that the items should be added and not multiplied. The clerk apologies and adds the items together and surprisingly total is still exactly $7.11. What are the exact costs of each item?
Problem 23
There are 12 people in a room. 6 people are wearing socks and 4 people are wearing shoes, 3 people are wearing both. How many people are in bare feet?
Problem 24
The figures below feature a logical relationship that is the same for each. Can you determine what number is represented by the question mark?
Problem 25
A zoo keeper puts six of his snakes into four cages. None of the cages is empty, and none of the cages contains an odd number of snakes. How is it possible?
Problem 26
What fraction of the Sierpinski triangle below is shaded?
Problem 27
Using the logic in the first three figures can you determine the missing number in the last figure?
Problem 28
Columns, rows and diagonals add up to 41. Fill in the missing numbers.
Problem 29
What number should replace the question mark in the sequence below?
5 12 14 7 10 ? 13 6 8 15
Problem 30
What fraction is represented by the shaded areas in the square below?
Problem 31
Can you move just 2 coins to make the triangle a square?
Problem 32
Can you determine the missing number in the last circle in the figures below?
Problem 33
A broken calculator does not display the digit 1. For example, if we type in the number 3131, only the number 33 is displayed, with no spaces. Mike typed a 6-digit number into that calculator, but only 2007 appeared on the display. How many different numbers could have Mike typed?
Problem 34
On a die the sum of the dots on opposite faces is always 7. Four such identical dice make up the figure in the picture. The dice are arranged such that the touching faces have the same number of dots. How many dots are on the face marked with the question mark?
Problem 35
Alex, Ben, Carl, and Daniel each participates in a different sport: karate, soccer, volleyball, and judo. Alex does not like sports played with a ball. Ben practices judo and often attends soccer games to watch his friend play. Which of the following statements is true?
A) Alex plays volleyball. B) Ben plays soccer. C) Carl plays volleyball. D) Daniel does karate. E) Alex does judo
Problem 36
See how many triangles you can draw using any three of the five points on the circle as vertices.
Problem 37
My brother and I were born in the same hour of the same day, of the same month, of the same year to the same biological mother and have the same biological father, but we are not twins. Why?
Problem 38
What is once in a minute, twice in a moment, but never in a second?
Problem 39
The five figures below use a similar logic to arrive at their interior numbers. Can you determine the missing number in the last figure?
Problem 40
Michael fell asleep on a plane halfway to his destination. He slept till he had as far to go as he went while he slept. How much of the whole trip was Michael sleeping?
Problem 41
Each boy: Mitch, Mark, Paul and Mathew has exactly one of four animals: a cat, a dog, a gold fish and a canary. Mark has an animal with fur, Mathew has an animal with four legs, Paul has a bird and Mitch and Mark don’t like cats. Which of the statements below is not true?
A) Mathew has a dog B) Paul has a canary C) Mitch has a gold fish D) Mathew has a cat E) Mark has a dog
Problem 42
The following 7 capital letters have a feature that other capital letters don’t have. Can you find out that feature? A B D O P Q P
Problem 43
There are 8 different magic puzzles with digits 1 through 9. Can you find all of them?
Problem 44
Without using any +, –, x or ÷ can you use three 3’s to make 3?
Problem 45
Can you express 15, 17, 19 and 21 using four 2’s?
Problem 46
How can 5 more than 11 be 4?
Problem 47
I am 5 digits long. I am divisible by 3 and 9, but not 6. My digits add up to 27. When my first and last digits are added, you will get a multiple of seven. My lowest digit is 3. If you add up my second and fourth digits, you get 10. No number is used more than once. On the left of the comma, the numbers from left to right decrease by 1. On the right of the comma, the numbers increase by 3.
Problem 48
In the first triangle rearrange numbers 1, 2, 3, 4, 5 and 6 so that the sum on each side of the triangle is 12. In the second triangle rearrange numbers 1, 2, 3, 4, 5 and 6 so that the sum on each side of the triangle is 10.
Problem 49
Write numbers 1 through 10 so each side of the pentagon has a sum of 14.
Problem 50
Each letter represents a digit. Same letters represent same digits. Find the digits that would make the equation true.
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