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This problem is from Ancient Rome.

$\\$ A rich senator died, leaving his wife pregnant. After the senator’s death it was found out that he left a property of 210 talents (an Ancient Roman currency) in his will as follows: “In the case of the birth of a son, give the boy two thirds of my property (i.e. 140 talents) and the other third (i.e. 70 talents) to the mother. In the case of the birth of a daughter, give the girl one third of my property (i.e. 70 talents) and the other two thirds (i.e. 140 talents) to the mother.”

$\\$ The senator’s widow gave birth to twins: one boy and one girl. This possibility was not foreseen by the late senator. How can the property be divided between three inheritors so that it is as close as possible to the instructions of the will?

In a class there are 50 children. Some of the children know all the letters except “h” and they miss this letter out when writing. The rest know all the letters except “c” which they also miss out. One day the teacher asked 10 of the pupils to write the word “cat”, 18 other pupils to write “hat” and the rest to write the word “chat”. The words “cat” and “hat” each ended up being written 15 times. How many of the pupils wrote their word correctly?

A six-digit number starts from the digit $1$. If this digit is relocated to the rightmost position, the number becomes $3$ times bigger. What is the number?

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?

If $R + RR = BOW$, what is the last digit of the number below?

$$F \times A \times I \times N \times T \times I \times N \times G$$

Replace the letters with digits in a way that makes the following sum as big as possible:

$$SEND + MORE + MONEY$$

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?

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.)

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.

Jane is playing the same game as Kate was playing in Example 3. Can she put together 5 cards with “+” signs and several “digit” cards to make an example on addition with the result equal to 2012

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.

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

$$\textrm{BAN}\times \textrm{G}=\textrm{BOOO}.$$ Different letters correspond to different digits, identical letters correspond to identical digits. The task is to solve the puzzle.

Do there exist three natural numbers such that neither of them divide each other, but each number divides the product of the other two?

The date $21.02.2012$ reads the same forwards and backwords (such numbers are called palindromes). Are there any more palindrome dates in the twenty first centuary?

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 t 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 rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Replace the letters with digits in a way that makes the following sum as big as possible:

SEND +MORE +MONEY

Looking back at her diary, Natasha noticed that in the date 17/02/2008 the sum of the first four numbers are equal to the sum of the last four. When will this coincidence happen for the last time in 2008?