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.)
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?
Do there exist three natural numbers such that neither of them divide each other, but each number divides the product of the other two?
Find all the solutions of the puzzle and prove there are no others. Different letters denote different digits, while the same letters correspond to the same digits. \[M+MEEE=BOOO.\]
In the following puzzle an example on multiplication is encrypted with the letters of Latin alphabet: \[{BAN}\times {G}= {BOOO}.\] Different letters correspond to different digits, identical letters correspond to identical digits. The task is to solve the puzzle.
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.
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
In the following puzzle an example on addition is encrypted with the letters of Latin alphabet: \[{I}+{HE}+{HE}+{HE}+{HE}+{HE}+{HE}+{HE}+{HE}={US}.\] Different letters correspond to different digits, identical letters correspond to identical digits.
(a) Find one solution to the puzzle.
(b) Find all solutions.
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.)