A pencil box contains pencils of different colours and different lengths. Show that it is possible to choose two pencils of both different colours and different lengths.
After proving there are no 3-digit numbers with the product of digits equal to 26 (see Example 1) Bella decided to find a 4-digit number with the product of digits equal to 98. Can she succeed in finding such a number?
Bella was encouraged by the fact that she fully understood the general concept about the existence of a number with given value of product of digits. Therefore, she started thinking about the following problems:
(a) Is there a 3-digit number with the sum of digits equal to 24?
(b) Is there a 4-digit number with the sum of digits equal to 37?
Solve these questions.
Once again consider the game from Example 2.
(a) Will you change your answer if the field is a rectangle?
(b) The rules are changed. Now you win if the sum of numbers in each row is greater than 100 and the sum of the numbers in each column is less than 100. Is it possible to win?
The text for this problem was originally typed in three different fonts and in three different colours. The original style is lost now, and Bella and Louise disagree on the following. Bella says that whatever the original font was it was always possible to choose three letters from the text such that all the three colours and all the three fonts were presented in that triple, and Louise does not think so. Who is right?
a) A bachelor student Peter haven’t slept properly for the last month. One of the reasons for that among many others was that every Monday at 3 p.m. he had a deadline for submitting his weekly calculus assignments. During the first month he counted six deadlines. Can it be the case or would you advise him to have more sleep?
(b) Once Peter checked the table with the assignment results he realized there were fewer Mondays in the last month. Is it possible there were only five Mondays?
(a) A picnic spot has a form of a 100 m\({}\times {}\)100 m square. Is it possible to partially cover it with non-intersecting square picnic blankets so that the total sum of their perimeters will be greater than 10,000 m?
(b) One sunny day almost every citizen came to the picnic spot from point (a). All of them brought square picnic blankets. In a local newspaper there was mentioned that the total area of grass covered with picnic blankets was greater than 20,000 m\(^2\). Do you think it was possible or did they make a mistake in their computations?
Scrooge McDuck has 100 golden coins on his office table. He wants to distribute them into 10 piles so that no two piles contain the same amount of coins. And moreover, no matter how you divide any of the piles into two smaller piles among the resulting 11 piles there will be two with the same amount of coins. Sounds impossible? Try to find a suitable example. Scrooge spent a while on working out this question, maybe he will even give you a penny.
There are 36 parcels weighing 1 kg, 2 kg, 3 kg, ..., 36 kg. Today only three cars are in service. Each car has a capacity of 12 parcels. Can one distribute all packages between the cars in such a way that each vehicle has the same total weight of parcels?
Can you cover the surface of a cube with 16 identical colourful rectangles? No overlappings are allowed and the cube has to be fully covered.