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.)
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 the letters with digits in a way that makes the following sum as big as possible: \[SEND + MORE + MONEY.\]
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?