Safin
Skilled
Come on rake your brains on these :
(Found them on a blog)
24 - Arithmetic - Hard
Using each of the numbers 1,3,4,6 create an expression which evaluates to 24. You may use any of the four standard arithmetic operations *,+,-,/ and parenthesis. You must use each of the numbers exactly once. You need not use any other numbers. It can also be done with 3,3,8,8.
Lockers - Number theory - Medium Easy
There are one hundred lockers in a row, all closed. Student number one walks by and opens every lockers. Student number two walks by and closes the even numbers. Student number three touches every third locker - locker numbers 3,6,9,12, etc. When he touches an open locker he closes it, when he touches a closed locker he opens in. Student number four changes every fourth locker. This continues for 100 students with student number n changing locker numbers n, 2n, 3n, etc. Which lockers are open and which are closed after this process?
The Monkey and the Coconuts - Number theory - Hard
5 sailors are shipwrecked on a deserted island. All they have to eat are coconuts, and they decide to divide them evenly tomorrow. They pile up all the coconuts in one big pile to sit overnight. That night, the first sailor wakes up and decides to take his share. He divides the coconuts into 5 equal piles, and has one left over which he throws to the monkey. He hides his pile, and puts back together the remaining 4 piles. Later that night, the second sailor wakes, makes 5 piles, again has one left over for the monkey and again hides his 'share'. In the course of the night each of the 5 sailors does this. So each has hidden some coconuts and the monkey has hidden five. The next morning, they divide the non-hidden coconuts into 5 piles but this time there are no extras for the monkey. How many coconuts were there?
Array Initialization - Programming - Medium
You've an algorithm which expects an n by n array initialized to zero. However, it typically only requires O(n) to run. You've memory to spare, but you'd like to avoid the n^2 initialization time each run. Describe a set of data-structures and provide the operations read, write, and clear. When used, they should look just like they were accessing a 2-d n by n array. However, all should have constant runtime (time independent of n).
Rectangles - Various - Very Hard
A rectangle is divided into smaller rectangles. Each of the smaller rectangles has the property that at least one of the sides has integer length. Show that the large rectangle has the same property. I've seen a particularly simple argument involving calculus, and found a not too long one involving number theory. I'm told that there are other proofs. While a bit abstract I think this one is cool for having multiple not obviously related solutions.
(Found them on a blog)
24 - Arithmetic - Hard
Using each of the numbers 1,3,4,6 create an expression which evaluates to 24. You may use any of the four standard arithmetic operations *,+,-,/ and parenthesis. You must use each of the numbers exactly once. You need not use any other numbers. It can also be done with 3,3,8,8.
Lockers - Number theory - Medium Easy
There are one hundred lockers in a row, all closed. Student number one walks by and opens every lockers. Student number two walks by and closes the even numbers. Student number three touches every third locker - locker numbers 3,6,9,12, etc. When he touches an open locker he closes it, when he touches a closed locker he opens in. Student number four changes every fourth locker. This continues for 100 students with student number n changing locker numbers n, 2n, 3n, etc. Which lockers are open and which are closed after this process?
The Monkey and the Coconuts - Number theory - Hard
5 sailors are shipwrecked on a deserted island. All they have to eat are coconuts, and they decide to divide them evenly tomorrow. They pile up all the coconuts in one big pile to sit overnight. That night, the first sailor wakes up and decides to take his share. He divides the coconuts into 5 equal piles, and has one left over which he throws to the monkey. He hides his pile, and puts back together the remaining 4 piles. Later that night, the second sailor wakes, makes 5 piles, again has one left over for the monkey and again hides his 'share'. In the course of the night each of the 5 sailors does this. So each has hidden some coconuts and the monkey has hidden five. The next morning, they divide the non-hidden coconuts into 5 piles but this time there are no extras for the monkey. How many coconuts were there?
Array Initialization - Programming - Medium
You've an algorithm which expects an n by n array initialized to zero. However, it typically only requires O(n) to run. You've memory to spare, but you'd like to avoid the n^2 initialization time each run. Describe a set of data-structures and provide the operations read, write, and clear. When used, they should look just like they were accessing a 2-d n by n array. However, all should have constant runtime (time independent of n).
Rectangles - Various - Very Hard
A rectangle is divided into smaller rectangles. Each of the smaller rectangles has the property that at least one of the sides has integer length. Show that the large rectangle has the same property. I've seen a particularly simple argument involving calculus, and found a not too long one involving number theory. I'm told that there are other proofs. While a bit abstract I think this one is cool for having multiple not obviously related solutions.
