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.
Notice that the square number 1089 \((=33^2)\) has two even and two odd digits in its decimal representation.
(a) Can you find a 6-digit square number with the same property (the number of odd digits equals the number of even digits)?
(b) What about such 100-digit square number?
Bella spent 10 minutes searching for a 3-digit number such that it has the product of it’s digits equal to 26. She examined all 3-digit numbers one by one. Do you think she missed a possible example or is it the case that there are simply no such 3-digit numbers?
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.
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?