So far we have learned that by considering all possible cases we can get a complete solution to a problem or prove that there is no solution at all (in case none of the possibilities leads us to an answer). Today we are going to improve our skills of finding what is called “narrow spots” in mathematical problems. By “narrow spot” we mean a feature of a given problem which can only have a few possibilities. Thus, starting with considering those possibilities may lead us to the shortest among all correct solutions. And having a clear solution without considering many cases is always nice!

Problems:

I don’t know how the figure below can be made of several $1\times5$ rectangles which do not overlap. I am willing to pay $\pounds 1$ if you show me a possible way of doing that which I have not seen before. What is the maximal amount of money a person can earn by solving this problem?

One gambler had a pair of dice. Rolling them was something that kept him concentrated. As a result of frequent usage all the numbers were wiped off from both of the dice. In January the gambler went through a rough patch and decided to take a break from gambling. He understood he could not rely only on his luck which has recently failed him. Therefore, our gambler started doing mathematical puzzles to master his mind. The first puzzle is to paint digits on each side of both dice (one digit per one side) in such a way that any natural number between 1 and 31 inclusive can be obtained by putting one dice next to the other. We do not allow the digit “6” to be used as the digit “9” and vice versa. Is there any solution to this problem?

Kate is playing the following game. She has 10 cards with digits “0”, “1”, “2”, …, “9” written on them and 5 cards with “+” signs. Can she put together 4 cards with “+” signs and several “digit” cards to make an example on addition with the result equal to 2012?$\\$ Note that by putting two (three, four, etc.) of the “digit” cards together Kate can obtain 2-digit (3-digit, 4-digit, etc.) numbers.

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) The second puzzle for our gambler is a bit similar to the first:

$\textit{“To paint digits on each side of both dice (one digit per one side) in such a way that}$ $\textit{any combination from 01 and 31 can be obtained by putting one dice next to the other.”}$

The digit “6” cannot be used as the digit “9” and vice versa. Is there any solution?\\ (b) What is the answer to (a) if we allow rotations (i.e. we allow the usage of “6” instead of “9” and vice versa)?

In the following puzzle an example on addition is encrypted with the letters of Latin alphabet:

$$\textrm{I}+\textrm{HE}+\textrm{HE}+\textrm{HE}+\textrm{HE}+\textrm{HE}+\textrm{HE}+\textrm{HE}+\textrm{HE}=\textrm{US}.$$ Different letters correspond to different digits, identical letters correspond to identical digits.$\\$ (a) Find one solution to the puzzle.$\\$ (b) Find all solutions.

(a) After building the garden the successful businesswoman had another idea in mind. She is keen to re-build the terrace in front of her country house. Now the goal is to plant nine sakura trees in such a way that one can count eight rows of trees each consisting of three trees (obviously, a tree can be counted in several rows). How the landscape gardener can satisfy this requirement?$\\$

(b) The neighbour of the businesswoman learned about her plans from the talk with the same landscape gardener and decided to outdo her with a similar but more complicated request. He is planning to plant nine sakura trees so that there can be found ten rows of three trees each. Is there a configuration of nine trees satisfying this condition?

A young mathematician had quite an odd dream last night. In his dream he was a knight on a $4\times4$ board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either $\\$

(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or$\\$

(b) he has passed exactly once through all the cells of the board. $\\$

For each possibility examine if it could happen or not.