Vectors -1, 0, 1
Consider all 1024 vectors in a 10-dimensional space with elements ±1. Show that if you change some of the elements of some of the vectors to 0, you can still choose a few vectors, such that their sum is equal to the 0-vector.
We do not know where this puzzle originated from. If you have any information, please let us know via email.
Consider all 1024 vectors in a 10-dimensional space with elements ±1. Show that if you change some of the elements of some of the vectors to 0, you can still choose a few vectors, such that their sum is equal to the 0-vector.
The sides of a rectangle have lengths which are odd numbers. The rectangle is split into smaller rectangles with sides which have integer lengths. Show that there is a small rectangle, such that all distances between its sides and the sides of the large rectangle have the same parity, i.e. they are all even or they are all odd.
A 1000 × 1004 rectangle is split into 1 × 1 squares. How many of these squares does the main diagonal of the large rectangle pass through?
You have 100 blue and 100 red points in the plane, no three of which lie on one line. Prove that you can connect all points in pairs of different colors
Alex and Bob are playing a game. They are taking turns drawing arrows over the segments of an infinite grid. Alex wins if he manages to create a closed loop, Bob wins if Alex does not win within the first 1000 moves. Who has a winning strategy if:
a) Alex starts first (easy)
b) Bob starts first (hard)
Remark: The loop can include arrows drawn both by Alex and Bob.
99 unique numbers between 1 and 100 are listed one by one, with 5 seconds pause between every two consecutive numbers. If you are not allowed to take any notes, what is the best way to figure out which is the missing number?
100 guests go to a party and some of them shake hands with each other. Show that there are two guests who handshake the same number of people.
A beetle is located in the center of a square carpet. The edges of the carpet are colored in red, green, blue, and yellow. Four spiders of the same colors are on the carpet’s corners. Each spider can only move on the edge with its matching color. Can the beetle escape the carpet and flee without encountering the spiders if it is 1.5 times slower than them?
With 12 matches you can easily create a shape with area 9 and a shape with area 5, as shown on the picture below. Can you rearrange the 12 matchsticks, so that they encompass an area of 4?
Remark: You should have only one resulting shape and no matches should be unused.
One programmer draws on a sheet of paper several circles in a line, representing coins, and puts his thumb on the first circle, covering the rest with his hand. Then he asks another programmer to guess how many different head-tail combinations are possible if someone flips all the (imaginary) coins on the paper. The second programmer, without knowing the number of circles, takes the pen and writes down a number. Then the first programmer lifts his hand and sees that the correct answer is written on the paper. How did the second programmer manage to do this?
Please confirm you want to block this member.
You will no longer be able to:
Please allow a few minutes for this process to complete.
Notifications