It is known that it is possible to cover the plane with any cube’s net. (You will see it in the film that will be shown at the end of this session). But Robinson, unfortunately, lived on an uninhabited island in the 19th century, and did not know about the film. Try to help him to figure out how to cover the plane with nets \(\#2\), \(\#6\), and \(\#8\) from the previous exercise.
In his twelfth year on the island Robinson Crusoe managed to tame a wolf, and used him as a guard dog for his goat. He used to tie the wolf with ropes to pegs and other ropes in such a way that there was always a safe place for the goat to escape. Subsequently, he studied the shapes the goat was grazing on the ground.
Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of a ring
Think of other shapes Robinson’s goat can graze without a wolf, or with a wolf tied nearby. What if Robinson managed to tame several wolves and used them as guard dogs? Can two tied wolves keep an untied goat in a triangle? Can you think of other shapes you can create with Robinson’s goat and wolves?
Find the smallest \(k\) such that \(k!\) (\(k!= k\times(k-1)\times \ldots \times 1\)) is divisible by \(2024\).
While studying numbers and its properites, Robinson came across a 3-digit prime number with the last digit being equal to the sum of the first two digits. What was the last digit of that number if among the number did not have any zeros among it’s digits?
When Robinson Crusoe’s friend and assistant named Friday learned about divisibility rules, he was so impressed that he proposed his own rule:
a number is divisible by 27 if the sum of it’s digits is divisible by 27.
Was he right?
One day Friday multiplied all the numbers from 1 to 100. The product appeared to be a pretty large number, and he added all the digits of that number to receive a new smaller number. Even then he did not think the number was small enough, and added all the digits again to receive a new number. He continued this process of adding all the digits of the newly obtained number again and again, until finally he received a one-digit number. Can you tell what number was it?
Can a \(300\)-digit number, with one hundred \(0\)s, one hundred \(1\)s, and one hundred \(2\)s among its digits, be a square number?
Robinson Crusoe’s friend Friday was looking at \(3\)-digit numbers with the same first and third digits. He soon noticed that such number is divisible by \(7\) if the sum of the second and the third digits is divisible by \(7\). Prove that he was right.
Prove the following divisibility rule by 37: divide the number starting from the right end of the number into blocks of three digits. Now, the original number is divisible by 37 if the sum of all three digit numbers obtained in this way is divisible by 37.
(It might be the case that the number of digits is not divisible by 3, and you cannot divide the original number into blocks of three digits. To overcome this problem we allow a block of three digits to start from 0, for example number 2345678 should be divided into blocks of three digits as 002, 345, and 678.)