Focus with the matrix. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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Focus with the matrix. Focus Secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

Prepare five coins and 20 paper chips. Ask someone to choose any of the numbers inscribed in the cells of the square (see picture). Put a coin on this number and close all other numbers that are in the same line and in the same row with the selected chips.

Now ask the same person to choose any of the numbers inscribed in the cells that are still not closed, put another coin on the chosen number, and cover the numbers in the same row and in the same column as the number chosen the second time with chips.

Repeat this procedure two more times, you will find that only one cell remains uncovered. Put the fifth coin on this cell.

If we now calculate the sum of the numbers covered with coins (recall that at first glance the numbers seem to be chosen at random), then it will be equal to 57. This is not accidental: no matter how many times you repeat the experiment, the sum will always be the same.

Focus secret:

The square is nothing more than the most common addition table, though compiled in a very intricate way. Such a table is built using two sets of numbers: 12, 1, 4, 18, 0 and 7, 0, 4, 9, 2. The sum of all these numbers is 57. By writing the numbers of the first set above the top row of the square, and the numbers of the second set on the left from the leftmost column, you will immediately understand how the numbers in the cells of the square are obtained.

Focus with matrix

So, the number in the upper left corner (standing at the intersection of the first row and the first column) is equal to the sum of the numbers 12 and 7. All other numbers are obtained in the same way: in order to find out which number should be entered in a particular cell, you just need to calculate the sum of the numbers in that row and that column, at the intersection of which the cell of interest to us is located.

In exactly the same way, you can build a magic square of any size with any numbers. How many cells are in the square and what numbers are chosen to build it, does not play any role.

The numbers in the source sets can be positive or negative, integer or fractional, rational or irrational. The resulting table will always have a magical property: by doing the above procedure with coins and chips, you will always get the sum of the numbers included in both original sets. In particular, in the case we have considered, one could take any eight numbers that add up to 57.

Now it is not difficult to understand the main idea of the focus. The number in any cell of the square is equal to the sum of some two numbers in the original sets. Putting a coin on the chosen number, you thereby, as it were, cross out these two numbers. Each new coin is placed at the intersection of another row with another column, so five coins correspond to the sum of the five pairs of initial numbers we have chosen, which, of course, is equal to the sum of all ten initial numbers.

One of the easiest ways to build an addition table using a square matrix is as follows. Let's write 1 in the upper left corner and continue numbering the cells from left to right with consecutive positive integers. A completed 4x4 matrix can be viewed as an addition table for two sets of numbers: 1, 2, 3, 4 and 0, 4, 8, 12. The sum of the numbers under the coins in such a matrix will always be 34.

The resulting amount, of course, depends on the size of the square. If the number of cells that fit along the side of the square is denoted by n, then the sum will be equal to (n3+n)/2.

Squares with odd n give the sum equal to the product of n and the number in the central cell. If the numbering of cells starts with a number a greater than 1 and continues in order, then the sum will be equal to ((n3+n)/2)*n(a-1).

It is interesting to note that the sum of the numbers in any column and in any row of the traditional magic square, made up of the same numerical elements, will be exactly the same.

Using the second formula, it is easy to find what the number in the upper left corner of a matrix of any size should be in order for it to give a predetermined sum. The following trick, which can be shown impromptu, makes a huge impression. When you ask someone to name any number greater than 30 (this will avoid negative numbers), you immediately draw a 4x4 matrix that will give the sum equal to the number just indicated! (For speed, instead of covering the numbers with coins, you can circle them, and cross out the rows and columns where the selected numbers intersect.)

To demonstrate this trick, you will have to do a single calculation (it is not difficult to do it mentally): subtract 30 from the named number, and divide the difference by 4. Let, for example, the number 43 be named. Subtracting 30, you get 13. Dividing it by 4, find the number 31/4. Entering 31/4 in the upper left corner of the 4x4 matrix and continuing further in order 41/4, 51/4, etc., you get a magic square with a sum equal to 43.

To further confuse the viewer, the numbers in the square should be rearranged. For example, the first number 31/4 can be entered in the cell in the third line, and the next three numbers (41/4, 51/4 and 61/4) can be placed in the same line, but in any order.

The next four numbers can be placed on any line, but in the same order in which you entered the first four numbers. The same must be done with the two remaining fours of numbers.

If you do not want to deal with fractional numbers, but still want to get the sum equal to 43, then the fraction 1/4 for all numbers can be discarded, and one can be added to the numbers in the top line (as a result of which the top line will be the numbers 16, 17, 18 and 19). In the same way, if the fractional part of the first number of the ball is 2/4, 2 would have to be added to the numbers in the top line, and if the fractional part was equal to 3/4 - 3.

Swapping rows and columns doesn't change the magic properties of the square, but it does make the matrix more mysterious than it really is.

Focus can also be shown with the multiplication table. In this case, the selected numbers should not be added, but multiplied. The resulting product is always equal to the product of the numbers with which the table is built.

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