I was very motivated to do such activity since young. Looking for patterns and solving the mystery.
How many different ways of making sure that the numbers placed in the boxes add up equally for both directions? How do you know you which number to place in the middle? Why does it have to be an odd number? Why does it have to be an even number?
I was impressed by the different ways of solving the problems and felt good unravelling the mystery behind.
I was once again reminded of this question which I had encountered many years backs, when I was still in my Primary school days.
“What is the sum of the first one hundred numbers beginning from one?”
1+2+3+ ... + 100 = ?
Today, I discovered how this question came about.
In the late 18^{th} century, a classroom teacher asked this question, the class, which was full of young kids, fell into complete silence. A few students were stunned by the seeminglyimpossible challenge and readily gave up; most students began scribbling on the paper, trying to add all the numbers one by one, from the very beginning. What a difficult question! They thought.
But there was one kid in the class did it differently. He thought about it for a few minutes, did some simple calculation, and raised his hand.
“I am finished,” the student said.
“How is it possible,” the teacher said to himself as he walked toward the student, “the problem would take one at least an hour to do!” Indeed, if he had solve the problem himself, he would just sum up all the one hundred numbers one by one as well – as a matter of fact, he presented the problem to the class just to kill some time. But after he examined his student’s answer, he was shocked.
The student was no other than Johann Carl Friedrich Gauss, one of the most famous and important mathematicians of all time.
So, how did young Gauss do the calculation?
First, he wrote the sum twice, one in an ordinary order and the other in a reverse order:
1+2+3... 99+100
100+99+ ... 3+2+1
By adding vertically, each pair of numbers adds up to 101:
1

+

2

+

3

+

. . .

+

98

+

99

+

100

100

+

99

+

98

+

. . .

+

3

+

2

+

1

101

+

101

+

101

+

. . .

+

101

+

101

+

101

Since there are 100 of these sums of 101, the total is 100 x 101 = 10,100. Because this sum 10,100 is twice the sum of numbers 1 through 100, we hence have:
1+2+3... 99+100 = 100 x 101 divide by 2 = 5050!!
This is also one of the way we used to solve our problem of finding the total number altogether. I am totally amazed by it.
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