Can a \(5\times5\) square checkerboard be covered by \(1\times2\) dominoes?
Cut an equilateral triangle into 4 smaller equilateral triangles. Then can another equilateral triangle be cut into 7 smaller equilateral triangles (triangles do not necessarily have to be identical)?
Consider another equilateral triangle. Is it possible to cut it into (a) 9; (b) 16; (c) 28; (d) 2; (e) 42 smaller equilateral triangles (which are not necessarily identical)?
(f) Kyle claims he can cut an equilateral triangle into any number of smaller (not necessarily identical) equilateral triangles if this number is either greater than 8 and divisible by 3, or greater than 3 and has remainder 1 when divided by 3. Prove or disprove Kyle’s statement.
(g)* Let \(n\) be a natural number greater than 5. Is it true one can cut an equilateral triangle into \(n\) smaller equilateral triangles?
a) What is the answer in case we are asked to split the figure below into \(1\times4\) rectangles instead of \(1\times5\) rectangles?
(b) In the context of Example 1 what is the answer in case we are asked to split the figure into \(1\times7\) rectangles instead of \(1\times5\) rectangles?
A hedge fund is intending to buy 50 computers and connect each of them with eight other computers with a cable. Please do not ask why they need to do that, that is a top secret never to be made public! A friend of mine said that it’s related to some cryptocurrency research, but you should immediately forget all I just told you; it would be unwise to spread rumours! Let’s go back to the mathematical part of this story and stop the unrelated talk. The question is, how many cables do they need?
At a party there are people dressed in either blue or green. Every person dressed in blue had a chance to dance with exactly \(7\) people in green, only once with each one. On the other hand, every person in green danced exactly with \(9\) people in blue, also only once with each. Were there more people dressed in blue or in green at the party?
Lady X has 3 different black skirts, and 5 different jackets – 3 blue, and 2 green. She also has 10 different hats – 6 blue and 4 green. Lady X’s outfit consists of a skirt, a jacket, and a hat of the matching colour.
In how many ways can the Lady choose her outfit?
Let us call a number super-odd if it is made of odd digits only. (For example, numbers \(5\), \(33\), \(13573\) are all super-odd.) How many \(3\)-digit super-odd numbers with all digits different are there?
Among 7 girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all 7 girls in a row such that the two girls wearing red shirts are not sitting adjacent to each other?
Gabby the Gnome has 3 cloaks of different colors: blue, green, and brown. He also has 5 different hats: 3 yellow and 2 red. Finally, he owns 6 different pairs of shoes: 2 yellow, and 4 red. Gabby is selecting an outfit: a cloak, a hat, and a pair of shoes. In how many ways can he do it if he wants the color of his shoes to match the color of the hat?