A young and ambitious software engineer is working on his own basic version of an intelligent personal assistant. The application can only answer closed questions (a closed question is a question that can be answered only ‘yes’ or ‘no’). He installs this application on three mobile devices and runs a set of tests. He discovers there is one unstable device. From time to time the application gives wrong answers, but you cannot really predict when. Being exhausted after unsuccessful attempts to find the mistake in his code, the software engineer goes to sleep. The next morning he cannot remember which device is not working properly. Taking into account that devices are connected to the same server (so normally working applications can detect which one is not always receiving the signal) explain how in two questions the engineer can determine the unstable device. One question is for one device only.
Assume you have a chance to play the following game. You need to put numbers in all cells of a \(10\times10\) table so that the sum of numbers in each column is positive and the sum of numbers in each row is negative. Once you put your numbers you cannot change them. You need to pay £1 if you want to play the game and the prize for completing the task is £100. Is it possible to win?
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.
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?
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?
a) In the context of Example 2 assume we have some number of parcels each weighing different amount of kilograms. We still have 3 identical cars of equal capacities (in numbers of packages) and we still want to distribute parcels in such a way that each car has the same total weight of parcels. Knowing that the number of parcels is not greater than 100 find the maximum and the minimum amounts of packages for which it is possible.
(b) Now we have 3 trucks so we do not really care about the sizes of parcels and their number. But yet we need to satisfy the condition of equal total weights of parcels in each vehicle. Can we do so if there are 27 packages weighing 1 kg, 2 kg, ..., 27 kg?
A battalion of soldiers was marching towards a captured city. Their progress was stopped by a wide river. Fortunately, close to the shore there were two boys sailing in a small boat. They escaped from the city and were eager to help the soldiers to cross the river. The only obstacle was that their boat could fit either two boys or one soldier. Taking into account one person was enough to handle that kind of boat (i.e. to sail from one shore to another) and the fact that on the next day the city was liberated so the boys could reunite with their families describe how the battalion was capable of crossing the river.