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Multiplication tables can be traced back to the Babylonians more than 4000 years ago. The earliest example of the decimal system appeared in China around 300 BC, where multiplication tables made of bamboo slips could calculate the product of integers and semi-integers less than 99.5; in addition, we can recognize the Pythagorean table mentioned by Nichomachus in his introduction to arithmetic (Introduction to Arithmetic) around 100 AD.

One of the earliest decimal multiplication tables appeared in China around 300 BC and was made of bamboo slips. Nowadays in school, the multiplication table is a tool for students to learn multiplication through rote memorization and quick memory exercises. Although some people think that mastering the multiplication table is an achievement in itself, it also lays a solid mathematical foundation for students. Let's dig deeper and reveal the mysteries hidden in the multiplication table from some interesting perspectives.

Number of triangles before explaining what a number of triangles is, let's take a look at this multiplication table and what we can do with it. The first row and first column of the table contain the numbers 1 to 10, while the other squares are filled with the product of the first number in the row and the first number in the column.

We add rows / columns 0 at the top and left of the table, which is still a multiplication table, just to make it easier for us to see some of the patterns below.

Now, let's paint the squares corresponding to multiples of 2 (all even numbers) blue. This means that all rows and columns corresponding to multiples of 2 are also blue, so we get a blue grid. The squares that are not in this blue grid are all white. (here we extend the table to the number 16 horizontally and vertically.)

Now, we paint all the squares with multiples of 3 blue. As before, we get a blue grid with rows and columns corresponding to multiples of 3. The remaining four white squares in the middle form a larger square (2 × 2 = 4):

If we paint all the squares with multiples of 4 blue, we can also get a blue grid. In this case, the place outside the blue grid forms a square with 3 × 3 squares, which are not entirely white because the square in the middle is blue. This happens because 4 is not a prime number.

In general, if you choose a positive integer k and represent the multiples of all k in the multiplication table in blue, you will get a corresponding blue grid, and the remaining two small squares will form a square. Whether k is a prime number determines whether these squares are pure white or contain some small blue squares.

This is interesting. Let's change to a k. The following picture shows the pattern we got from Kraft 6 (you can easily imagine the pattern of Kwon 5, because 5 is a prime number).

Let's see how the number of triangles appears in the figure. The number of triangles is a number that can be represented by a pattern of points arranged in an equilateral triangle with the same number of points on each side and the same spacing.

For example:

The first triangle is 1, the second is 1 / 2 / 3, the third is 1 / 2 / 3 / 6, the fourth is 1 / 2 / 3 / 4 / 10, and so on. In general, the nth triangle number Tn is the sum of the first number 1 to n:

How can we find these magic numbers in the box of the multiplication table? First of all, let's take a look at the multiplication table again, where the square corresponding to the multiple of 3 is blue. We ignore the multiplication table where blue is a multiple of 2 because mathematicians think it's trivial: it doesn't make any sense. The first white square in the multiplication table after a multiple of 3 is painted blue looks like this:

Add up the numbers in this white square to get:

9 is not a triangle number, but it is the square of a triangle number. To be exact, it is the square of the number of the second triangle T2.

Now, let's take a look at the first white square after painting the multiplication table with a multiple of 4 corresponding to a small square blue:

Add up the numbers in this square (including the numbers in the middle blue square) to get the result:

In this case, the sum is equal to the square of the third triangle.

It won't be long before you find the same pattern in Knights 5 and 6.

When kicking 5, the sum of the numbers in the first square:

When kicking 6:

Is this a general rule?

Is it true that we paint any multiple of k blue? If so, after the sum of all the numbers in the first square enclosed after the multiple of k in the multiplication table is painted blue, the number Tk-1 of the first triangle can be obtained.

Let's see if this is correct. In the multiplication table, we will see that the numbers in the first row of squares are:

The second line is multiplied by these numbers by 2:

The third line is multiplied by the number in the first row by 3:

Continue one by one in this way until the last line of the square: multiply the number of the first row by (kmur1):

Then add up the numbers in these lines:

Put forward (1-2-3 +... + kMur1), the formula becomes:

As mentioned above:

Therefore, we prove that the sum of all the numbers in the first large square Tk-12 is equal to the square of the number of triangles.

Square in the ocean of integers, the red number on the principal diagonal of the multiplication table (from the northwest corner to the southeast corner) is obviously the square-the 2 power of the integer.

Not only the number of triangles but also the square number can be found in the multiplication table. As we know in the previous introduction, the multiplication table fills the multiple of k with blue, and the sum of the numbers in the square surrounded by these blue squares is related to the number of triangles. The sum of the numbers in the square is equal to (2m-1) (2n-1) Tk-12, where m and n represent the number of squares calculated from the top and the left, respectively, and Tk-1 is the first number of triangles.

We can see that the sum of the square lattice surrounded by blue multiples on the principal diagonal (from the northwest corner to the southeast corner) is also square. This can be easily proved from the original summation formula of the article, because the vertical and horizontal positions are the same, we only use m in the formula:

Split squares if we delve into the square structures of other different sizes and positions in the multiplication table, we can find more squares. Square lattices based on principal diagonals always seem to produce squares, which are closely related to the sum of column and row indicators shared by the selected squares.

The square number 22 square is obtained from the single square in row 2 and column 2 (orange part). There are four squares (red) where rows 3 and 4 overlap with columns 3 and 4, and the numbers in the four squares are added together to get (3'4) 2'49; while rows 5, 6, 7 and 5, 6, 7 overlap nine squares (green). Add the numbers of these nine squares together to get (5'6'7) 2 '324.

The multiplication table has row metrics on the left and column metrics on the top.

This also seems to be true when a square is generated by the intersection of discontiguous rows and columns. If we take the intersection of rows 1, 4, 8 and columns 1, 4, 8, the sum of the median numbers in the (discrete) box is: (1: 4: 8) 2: 169.

For the squares defined by the three integers a, b, c in the multiplication table, the formula suitable for all three numbers can be obtained by mathematical operation. In the above example, the sum of the numbers in the box is:

More generally, there are:

By summing the numbers in the intersection grid of the same row index (a, b, c) and the corresponding column index (a, b, c), the square of the row / column index sum is given. Can this be extended to four numbers, five numbers, or even more?

Square of square and square of cube

Based on this knowledge, we can find some special patterns. For example, let's look at rows with consecutive odd numbers as row and column metrics, and you will quickly find that the sum of consecutive odd numbers (starting with 1) is equal to a square number.

Because the sum of consecutive odd numbers is a square number, the sum of row / column metrics corresponding to continuous odd numbers is a square number. Then the square of the sum of row / column indicators will be the square of a number: that is, the fourth power of a number. Therefore, we can use this special lattice form to get positive integers to the fourth power from the multiplication table.

Summing the blue square at the intersection of a continuous odd row and a continuous odd sequence results in a number to the fourth power.

We can use another interesting conclusion that a cubic number (to the third power of a number) can be written as a continuous odd sum. For example, 13, 1, 23, 8, 3, 5, and 33, 27, 7, 9, 11. Therefore, if we choose the square lattice of the intersection of these consecutive odd rows and odd columns, the sum of the numbers in these square lattices will be the square of a cube, that is, the sixth power of a number. The green squares below are the intersection of rows 3 and 5 and columns 3 and 5, and their sum is (3-5) 2 = (23) 2-26. The yellow square is the intersection of rows 7, 9, 11 and columns 7, 9, 11, and their sum is (7: 9: 11) 2 = (33) 2: 36.

Math teachers are always looking for new ways to introduce the concepts of multiplication, exponent and algebra. If we jump out of the mindset, we will find that the multiplication table is not just a tool for memorizing the multiplication table. If we choose to dive into the deep blue water, we will find many mathematical treasures at the bottom of her sea.

Original text link:

Https://plus.maths.org/content/powers-multiplication-table

Https://plus.maths.org/content/triangular-patterns

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop). Authors: Tony Foster, Sai Venkatesh, Zoheir Barka & the Plus team, translator: Crunc, revision: zhenni

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