You hear your name and it should appear on the board – these are not necessarily the best memories of school. Fortunately, I survived the helpless guesswork in front of the class. But maybe you would solve the task on the board with flying colors?

It’s the squares of numbers 1 .^{2} up to 100^{2}. Find the sum of all even square numbers and subtract the sum of all odd square numbers.

What is the result of the calculation?

the result is **5050**. This is exactly the sum of all numbers from 1 to 100.

Finding the solution is not particularly difficult when we have binomial formula

**a**^{2 }**– B**^{2}** = (a – b) * (a + b) **

use.

We rearrange the squares – in descending order of size – and add the corresponding sign in each case – that is, plus or minus:

Total = 100^{2} – 99^{2} +98^{2} – 97^{2} + … + 2^{2} – 1^{2}

We now rewrite the differences between two square numbers using the binomial formula:

Total = (100-99) * (100 + 99) + (98-97) * (98 + 97) + … + (2-1) * (2 + 1)

Since adjacent numbers always differ from exactly 1, we can simply omit the expressions (100-99), (98-97) and so on. This greatly simplifies the calculation:

Total = 100 + 99 + 98 + 97 + … + 2 + 1

We don’t have to add the numbers from 1 to 100 in our heads, we use the trick with which the mathematician Carl Friedrich Gauss once impressed his teacher as a student. To do this, we simply rearrange the numbers into 50 pairs, totaling 101 for each pair:

Sum = (100 + 1) + (99 + 2) + … + (51 + 50)

Total = 50 * 101 = 5050

*If you missed a puzzle from the past few weeks, here are the last ten episodes:*

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