Problems

Draw $6$ points on a plane and join some of them with edges so that every point is joined with exactly $4$ other points.

There are $15$ cities in Wonderland, a foreigner was told that every city is connected with at least seven others by a road. Is this enough information to guarantee that he can travel from any city to any other city by going down one or maybe two roads?

Replace letters with digits to maximize the expression $$NO+MORE+MATH.$$ (In this, and all similar problems in the set, same letters stand for identical digits and different letters stand for different digits.)

Jane wrote a number on the board. Then, she looked at it and she noticed it lacks her favourite digit: $5$. So she wrote $5$ at the end of it. She then realized the new number is larger than the original one by exactly $1661$. What is the number written on the board?

Find the biggest 6-digit number such that each of its digits, except for the last two, is equal to the sum of its two right neighbours.

Replace the letters with digits in a way that makes the following sum as big as possible: $$SEND+MORE+MONEY.$$

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit $5$. Jane then decided to include it, but the number was written too close to the edge, so she decided to fit the $5$ in between the two digits. She noticed that the resulting number is $11$ times larger than the original. What is the sum of digits of the new number?

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the first two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the “6-digit number” restriction.)

If $R+RR=BOW$, what is the last digit of the number below? $$F\times A\times I\times N\times T\times I\times N\times G.$$

Shmerlin the magician found the door to the Cave of Wisdom. The door is guarded by Drago the Math Dragon, and also locked with a 4-digit lock. Drago agrees to put Shmerlin to the test: Shmerlin has to choose four integer numbers: $x,y,z$ and $w$, and the dragon will tell him the value of $A\times x+B\times y+C\times z+D\times w$, where $A,B,C$ and $D$ are the four secret digits that open the lock. After that, Shmerlin should come up with a guess of the secret digits. If the guess is correct, Drago will let the magician into the cave. Otherwise, Shmerlin will perish. Does Shmerlin have a way to succeed?