New Year Puzzle:In amagic square, the sum of numbers on each row, each column, and each diagonal is the same. Fill in the numbers in the following squares to make them become magic squares.

### Merry Christmas and Happy New Year

### Which is smaller? which is larger?

Today let us look at the following two tricky questions:

**Question 1:**Among these two numbers 4444 x 55555 and 5555 x 44444, which one is smaller? which one is larger?

**Question 2:**How about 5000 x 5000 and 4999 x 5001, which one is smaller? which one is larger?

### Vianney's awesome square!

- each entry is a square number: $$1=1^2, \quad25=5^2, \quad 49=7^2$$
- the sum of numbers on each row, each column, and each diagonal is the same.

### Pythagorean Identity

This year we will start a blog series about mathematical identities. Let us begin today with the Pythagorean Identity $$ a^2 + b^2 = c^2 $$

### Fibonacci numbers and continued fractions

Fibonacci is probably the most well known sequence in mathematics. Today, we will see how Fibonacci numbers can be used to construct beautiful patterns called

*continued fractions*.

### Divide a line segment by compass

Using compass and straightedge, we can construct the midpoint of a line segment, and we can easily divide a line segment into, say three equal parts. The question is, is it possible to do these constructions

**with just the compass**.

The answer is "yes"! Indeed, it is possible to construct the midpoint of a line segment, and it is possible divide a line segment into a number of equal parts by using the

**compass alone**. How amazing is that?! Today, we will look at these constructions!

### Construction by compass alone

Normally in geometric construction problems, we use straightedge and compass. Today, we will look at an unusual type of construction problems where we are only allowed to use compass.

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