What a Naughty Boy Can Teach Us About Adding

Over 200 years ago, an unruly first-grader continued to misbehave in class. One day his teacher had finally had enough of the boy’s antics and banished him to the corner. To atone for his rambunctiousness, the boy was forced to add the numbers 1-100. Only then would he be able to return to the fold.

Having consigned him to the corner, the teacher was livid when a few moments later the boy stood up and walked back to his desk. The gall of this fractious boy, she probably thought, and stomped over to his desk ready to mete out a punishment far worse then arithmetic torture.

When she stood over him she saw he’d scrawled the number  5,050 on a piece of paper.

“What is this,” she demanded.

The little boy looked up at her smugly and replied, “1 + 2 + 3… all the way up until a hundred.”

Had the teacher not known the answer, she may have gone so far as to flagellate the impertinent lad, but slowly she recalled that 5,050 was indeed the answer.

So how did this little boy sum all these number in such a short time?

Well, he took the first and last number in the series, 1 and 100, and added them, getting 101. He then noticed he’d get the same sum when he added 2 + 99. In this manner, he continued pairing off numbers, 3 + 98, 4 + 97, seeing that the sum remained the same, 101. He then asked himself how many pairs of numbers are there in 100 numbers. The answer 100/2 or 50. So he thought, a surge of victory flowing through him, if each pair equals 101 and there are 50 pairs: 50 x 101 = 5,050.

In much the same way, we can add series of numbers by pairing the first and last term and then finding the number of pairs in the series. Remember to find the number of numbers in a consecutive sequence you need to subtract the first from the last and add 1. Try the question below:

What is the sum of the consecutive integers 50 to 150?

(A) 5,050

(B) 8,500

(C) 10,100

(D) 10, 210

(E) 12,800

(First + Last) x (# of numbers in series) =


(50 + 150) x (150 – 50 + 1) /2 = 10,100. Answer C

Oh, one more thing: while I took liberties on this story, it is by no means apocryphal. The little boy turns out to be Karl Friedrich Gauss, arguably the greatest German mathematician of all-time.

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