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?
A six-digit number starts with the digit \(1\). If this digit is relocated to the rightmost position, the number becomes \(3\) times bigger. What is the number?
Solve: \(HE \times HE = SHE\).
Prove that the number of all arrangements of the largest possible amount of peaceful bishops (figures that move on diagonals and don’t threaten each other) on the \(8\times 8\) chessboard is an exact square.
In a bookcase, there are four volumes of the collected works of Astrid Lindgren, with each volume containing 200 pages. A worm who lives on this bookshelf has gnawed its way from the first page of the first volume to the last page of the fourth volume. Through how many pages has the worm gnawed its way through?
Deep in a forest there is a small town of talking animals. Elephant, Crocodile, Rabbit, Monkey, Bear, Heron and Fox are friends. They each have a landline telephone and each two telephones are connected by a wire. How many wires were required?