A simple magic in Mathematics

SP different life simple mathematics C method friend

Draw 11 small squares side by side on a piece of paper. Call your friend back to you (make sure you don't see anything on the paper) and fill in the first two squares with numbers ranging from 1 to 10. Start from the third square and fill in the sum of the numbers in the first two squares in each square. Let your friends always calculate the number of 10th squares. If your friends fill in 7 and 3 squares at the beginning, the number in the first 10 squares should be

7 3 10 13 23 36 59 95 154 249

Now, ask your friend to report the number in the second square. You only need to press a few keys on the calculator to tell the number in the second square. Your friends will be surprised to find that the result is exactly the same as your prediction by calculating the number of 11th squares! This is strange. If you don't know the number of the first two numbers, you only know the size of the number 10th and the size of the number 9th, how can you guess the value of the number 11th? Magic secrets: divide by 0.618

In fact, it is very easy to estimate the number of 10th by the number of 11th. All you need to do is divide the number of 10th by 0.618. The result is rounded to 11th. In the above example, because 249 records 0. 618 = 402. 913... ≈ 403, You can confidently conclude that 11th is 403. In fact, the sum of 154 and 249 is actually equal to 403. Change the number in the first two squares. The conclusion is still true:

2 9 11 20 31 52 82 133 215

As you can see, the number of 11th should be 215 + 348 = 563, and the number of 348 divided by 0.618 is equal to 563. 107..., which is surprisingly consistent with the actual results. What exactly is this? Magic principle: Inspiration from solution allocation

Assume that the two numbers that your good friend wrote on paper are a and B. The numbers in these 11 squares are:

A B A + 2B 2a + 3B 3A + 5B 5A + 8B 8A + 13B 13a + 21B 21a + 34B 34a + 55B

Next, we only need to explain that the result of dividing 21a + 34B by 34a + 55B is very close to 0.618.

Let's consider another seemingly unrelated life tips: the two cups of brine are mixed with different concentrations, and the allocated Brine Concentration must be between the original two cups of Brine Concentration. In other words, if the concentration of one cup of Brine is A/B and the concentration of the other cup of Brine is C/d, then (A + C)/(B + D) it must be between A/B and C/D.

Therefore, (21a + 34B)/(34a + 55B) must be between 21a/34a and 34B/55B. 21a/34a = 21/34 ≈ 0.6176, 34B/55B = 34/55 ≈ 0.6182, no matter how much A and B are, (21a + 34B)/(34a + 55B) both are clamped between 0.6176 and 0.6182. If neither a nor B is large, it is quite reliable to divide the value of 21a + 34B by 0.618.

Some readers may have discovered that 0.618 is not another number, but a mysterious golden split. The coefficients 1, 1, 2, 3, 5, 8, 13, 21 appear in the table above, 34, 55 ,... It is the legendary Fibonacci series. The two concepts of the most mysterious colors in arithmetic are intertwined here. It seems that this simple little magic is not simple.