## The 7-Digits Puzzle (Puzzle7)

Use each digit from 1 to 7 exactly once, and place them into the circles of Figure 1 in such a way that the sum along each of the five lines is the same.

Use each digit from 1 to 7 exactly once, and place them into the circles of Figure 1 in such a way that the sum along each of the five lines is the same.

This model draws a clock. It is the template for the following simple puzzle: “Divide the clock with a straight cut into two parts such that the sum of the numbers in both parts are equal ?”. Can you see the solution at once?

“One day a customer arrived at a 7-Eleven shop and selected four items. He then approached the counter to pay for these items. The salesman took out his calculator, pressed a few buttons, and said: “The total price is $7.11.” “Why?” said the customer, who was amazed by this coincidence. “Do I have to pay $7.11 because the name of the shop is 7-Eleven?” “Of course not!” replied the salesman. “I have multiplied the prices of these four items and I have given you the result!” “Why did you multiply these numbers?” asked the surprised customer. “You should have added them to get the total price!” “Oh, yes, I’m sorry,” the salesman said. “I have a terrible headache and pressed the wrong button!” Then the salesman repeated all the calculations (i.e., he added the prices of these four items), but to both of their great surprise, the total was still $7.11. What are the prices of these four items? Believe it or not, there is a unique solution to this problem…” (This puzzle is from [3], Puzzle 12-6.)

A customer entered a shop and asked to change a single dollar for coins. The cashier said: “I am sorry, but I can not change it with the coins I have.” The customer then asked for change of a half dollar, but the cashier replied: “I am sorry, but I can not change it with the coins I have.” The customer asked for a change of a quarter, then a dime, then a nickel, but for all these requests the cashier kept replying: “I am sorry, but I can not change it with the coins I have.” Finally, the customer got upset and asked: “Do you have any coins at all?” The cashier replied: “Yes … I have $ 1.15 in coins.” The question is: what coins were in the cash register? (This puzzle is from [3], Puzzle 5-5.)

A father wants to pass on his valuable coin collection to his 7 children. The oldest child gets half of all coins and in addition half a coin. The remaining coins are distributed in the same manner among the younger children. How many coins does he distribute?

The puzzle presented in math21 gives rise to a number of interesting other puzzles, if we generalize it slightly: define a grid of dimension IxI within I > 4. Place exactly d (with d < I) coins on each row and each column. Furthermore, the coins should be placed in a symmetrical way with respect to one diagonal.

This problem gives a formulation for question 2 in model math21. Take 16 coins and put them in four rows of four each. Remove a certain number z of coins in order to make the number of coins in each row and each column divisible by 3.

Take 16 coins and put them on a 4 × 4 grid such that every cell is covered by a coin. Remove 6 coins leaving an even number of coins in each row and in each column (see [1] and [2]).

Each of the cells in a 4×4 square grid can be in one of two states, white or black. The starting configuration is given as a parameter R (1 means white, 0 means black).

Each of the cells in a 5×5 square grid can be in one of two states: white or black. Initially, all cells are white. If the player clicks in a cell then that cell and all 4 vertical and horizontal neighbors will toggle between the two states. Each mouse click constitutes one move and the objective of the puzzle is to turn all 25 cells black with the smallest number of clicks. (see [1]).