07/24/2013
Math isn't hard!
Different characters stand for different non-zero digit numbers in the following equality:
MATH + ISNT = HARD ,
where I=1 and N=8. Find out the values of all the characters.
06/02/2013
Four integers
Find all positive integers a,b,c,d such that 1/a+1/b+1/c+1/d=1.
05/18/2013
Clock hands
When do hour hand, minute hand and second hand on a clock meet all together?
04/18/2013
Adding up astronomical figure to a single-digit
Given any number, e.g., 123456789, add one or more "+" between its cyphers and calculate the sum. For the given example, we could do it like this: 1+234+5678+9=5922. We can continue to do the same for 5922. Prove that, for any number, it is possible in 4 steps to get a single-digit number.
Solution here (mirror).
03/29/2013
Chicken Nuggets
You can buy Chicken McNuggets in 9 and 20 piece packages. What is the largest amount you can never buy assuming that you do not eat or take away any McNuggets?
Solution.
03/14/2013
Look and Say Sequence
What is the next number of the sequence: 2, 12, 1112, 3112, 132112,...?
Answer.
02/11/2013
Easy proof?
Prove that there exist two positive irrational numbers r,t such that r^t is rational.
01/28/2013
World's Hardest Easy Geometry Problems
Problems (mirror, solutions)
12/27/2012
Pingpong Balls
All the pingpong balls look the same, only one counterfeit has different weight from the others.
Given a scale, weigh 3 times:
1) What is the biggest number of balls, from which you can find out the counterfeit one, if you know the counterfeit ball is either lighter or heavier.
2) What is the biggest number of balls, from which you can find out the counterfeit one, if you do not know the counterfeit ball is either lighter or heavier.
12/24/2012
More Counting
1. Mom is distributing apple, banana, orange, pear and mango one each to her four kids Adam, Bo, Obama and Peidi. Each kid will get at least one fruit. How many ways can be distributed?
2. A teacher is trying to divide 20 students into 2 groups. Suppose each group will have at least one student. How many ways can the teacher go with?
3. A lock has 9 buttons numbered 1-9. The lock is opened by pushing two buttons simultaneously and then pushing two buttons in sequence. How many combinations are possible?
4. In how many different ways can 5 men and 6 women be placed into five groups of two people and one group of three people if there must be at least one man and one woman in each group?
11/28/2012
A Bizarre Problem
A teacher whispers positive integer A to Aaron, B to Bob and C to Charles. The students do not know each others' numbers but they are told the sum of their numbers is 14. Aaron says, "I know that Bob and Charles have different numbers". Then Bob says, "I already knew that all three of our numbers were different." Finally, Charles declares that he already knew all these three numbers. What are they?
10/20/2012
Counting Triplets (to my 7th-8th grader students)
How many triplets (x,y,z) will make 12-digit number 123x456y789z divisible by 11?
10/18/2012
21! (to my 7th grader students)
Can you determine the ten thousand's digit of 21! without much calculations?
10/11/2011
Forever 21 (to my friends at Forever 21 Volleyball Club)
Consider the expression (21 /'s)
21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21 .
If you are allowed to put as many parentheses as you want, how many different values can be obtained for this expression? Solution.
10/02/2012
Northeast Bound
You are at the origin (0,0). You are asked to jump to (4,4). If you are only allowed to jump 1,2,3 units either eastward or northward, how many ways can you take to arrive at the destination?
09/26/2012
Rational to Decimal
It's being taught that every rational number can be written into terminating or repeating decimal.
What is the reason behind it?
09/03/2012
An Equilateral Dilemma
An equilateral triangle DEF with vertices D,E,F on sides AB,AC,BC of a triangle ABC respectively satisfies AD=BF=CE, prove or disprove that triangle ABC is an equilateral triangle.
This problem seems to originate from IBM.
A proof can be found here (a mirror copy here).
Note: The original proof had a flaw. I contacted Brent and he corrected it on 09/11/2012.
08/24/2012
24 points
PEMDAS game: use parentheses, multiplications, divisions, additions and subtractions, can you turn the following four numbers, 8, 8, 3 and 23 (used once) into 24?
Check it here.
06/18/2012
9-fold number
Is there a number that if you move the last digit to the very beginning, the resulting number is equal to 9 times of the original number? Answer.
Note: My friend Zhang found a very elementary solution to the above problem, let me know if you are interested.
What to take home?
1. Fermat's little theorem
2. Powerful usage of www.wolframalpha.com
3. A bit knowledge on Parasitic number/Dyson number
Math isn't hard!
Different characters stand for different non-zero digit numbers in the following equality:
MATH + ISNT = HARD ,
where I=1 and N=8. Find out the values of all the characters.
06/02/2013
Four integers
Find all positive integers a,b,c,d such that 1/a+1/b+1/c+1/d=1.
05/18/2013
Clock hands
When do hour hand, minute hand and second hand on a clock meet all together?
04/18/2013
Adding up astronomical figure to a single-digit
Given any number, e.g., 123456789, add one or more "+" between its cyphers and calculate the sum. For the given example, we could do it like this: 1+234+5678+9=5922. We can continue to do the same for 5922. Prove that, for any number, it is possible in 4 steps to get a single-digit number.
Solution here (mirror).
03/29/2013
Chicken Nuggets
You can buy Chicken McNuggets in 9 and 20 piece packages. What is the largest amount you can never buy assuming that you do not eat or take away any McNuggets?
Solution.
03/14/2013
Look and Say Sequence
What is the next number of the sequence: 2, 12, 1112, 3112, 132112,...?
Answer.
02/11/2013
Easy proof?
Prove that there exist two positive irrational numbers r,t such that r^t is rational.
01/28/2013
World's Hardest Easy Geometry Problems
Problems (mirror, solutions)
12/27/2012
Pingpong Balls
All the pingpong balls look the same, only one counterfeit has different weight from the others.
Given a scale, weigh 3 times:
1) What is the biggest number of balls, from which you can find out the counterfeit one, if you know the counterfeit ball is either lighter or heavier.
2) What is the biggest number of balls, from which you can find out the counterfeit one, if you do not know the counterfeit ball is either lighter or heavier.
12/24/2012
More Counting
1. Mom is distributing apple, banana, orange, pear and mango one each to her four kids Adam, Bo, Obama and Peidi. Each kid will get at least one fruit. How many ways can be distributed?
2. A teacher is trying to divide 20 students into 2 groups. Suppose each group will have at least one student. How many ways can the teacher go with?
3. A lock has 9 buttons numbered 1-9. The lock is opened by pushing two buttons simultaneously and then pushing two buttons in sequence. How many combinations are possible?
4. In how many different ways can 5 men and 6 women be placed into five groups of two people and one group of three people if there must be at least one man and one woman in each group?
11/28/2012
A Bizarre Problem
A teacher whispers positive integer A to Aaron, B to Bob and C to Charles. The students do not know each others' numbers but they are told the sum of their numbers is 14. Aaron says, "I know that Bob and Charles have different numbers". Then Bob says, "I already knew that all three of our numbers were different." Finally, Charles declares that he already knew all these three numbers. What are they?
10/20/2012
Counting Triplets (to my 7th-8th grader students)
How many triplets (x,y,z) will make 12-digit number 123x456y789z divisible by 11?
10/18/2012
21! (to my 7th grader students)
Can you determine the ten thousand's digit of 21! without much calculations?
10/11/2011
Forever 21 (to my friends at Forever 21 Volleyball Club)
Consider the expression (21 /'s)
21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21 .
If you are allowed to put as many parentheses as you want, how many different values can be obtained for this expression? Solution.
10/02/2012
Northeast Bound
You are at the origin (0,0). You are asked to jump to (4,4). If you are only allowed to jump 1,2,3 units either eastward or northward, how many ways can you take to arrive at the destination?
09/26/2012
Rational to Decimal
It's being taught that every rational number can be written into terminating or repeating decimal.
What is the reason behind it?
09/03/2012
An Equilateral Dilemma
An equilateral triangle DEF with vertices D,E,F on sides AB,AC,BC of a triangle ABC respectively satisfies AD=BF=CE, prove or disprove that triangle ABC is an equilateral triangle.
This problem seems to originate from IBM.
A proof can be found here (a mirror copy here).
Note: The original proof had a flaw. I contacted Brent and he corrected it on 09/11/2012.
08/24/2012
24 points
PEMDAS game: use parentheses, multiplications, divisions, additions and subtractions, can you turn the following four numbers, 8, 8, 3 and 23 (used once) into 24?
Check it here.
06/18/2012
9-fold number
Is there a number that if you move the last digit to the very beginning, the resulting number is equal to 9 times of the original number? Answer.
Note: My friend Zhang found a very elementary solution to the above problem, let me know if you are interested.
What to take home?
1. Fermat's little theorem
2. Powerful usage of www.wolframalpha.com
3. A bit knowledge on Parasitic number/Dyson number