For many people it seems that the prospect of even the simplest mental arithmetic can cause a form of brain paralysis, partly from a fear of maths and partly from the ease with which you can grab a calculator. But the ability to calmly analyse a situation and work out the quickest and easiest way of resolving a problem can be a great asset in many walks of life, not just mathematics.

For the International Maths Year in 2000 I wrote an educational maths musical entitled, “Tales of Maths and Legends” that we toured around UK theatres, playing to huge audiences. One of the show’s songs featured one of the greatest mathematicians of all time, Carl Friedrich Gauss:

Carl Friedrich Gauss, when only nine
He did a sum in record time,
“Please do this sum,” his teacher said,
“Add the numbers from One up to a Hundred”

We’d then asked the audience, “We want you to add all the numbers from 1 to 100. Ready? Steady? Go.....” We paused for just a few seconds and then asked, “Have you done it?”

Of course they hadn’t done it. Adding 1 + 2 + 3 etc, right up to 100 would take ages, even with a calculator. But Gauss managed it, when only young, so there must be a trick to it. So what is the trick? It’s really very simple.

Imagine the numbers from 1 to 100 written out in a long line. The two end numbers are 1 and 100, which added together make 101. But the next two numbers in from each end, 2 and 99, also add together to make 101, and the third from each end, 3 and 98 also add up to make 101, and so on.

How many pairs of number can you make from 100 numbers? 100 ÷ 2 = 50. So there are 50 pairs of 101. So, writing on a huge board, we sang to the music:

The hundred numbers added up, well this is how you go,
You make a pair of first and last, and write it down like so
Then multiply by 50 pairs, it makes Five Oh, Five Oh,
It’s one of the Tales of Maths and Legend.

Or, without the song, 101 x 50 = 5050

So in no more than a couple of minutes, we had set what some people can see as an almost impossible question and shown the answer to be simplicity itself, once you are thinking mathematically. Mathematical thinking can and does empower you.

Alright for Gauss you might think, but he was a mathematical genius. And sadly, the Gauss story isn’t quite accurate. Gauss did answer a similar question when he was nine-years-old back in 1876. He did write the answer on his school slate (before paper was cheap enough to be wasted on school work) without showing any working. But the actual question, which most probably came after the teacher had explained the 1 to 100 puzzle, was set along these lines.

“You have the number 54217, then the number 54234, then 54251, and so on. What is the pattern?” asks the teacher.
“Please sir, each one increases by 17.”
“Correct, that boy,” said the teacher. “Now, I want you to give me, before you go home, the total of the first 100 numbers in this series added together, starting with 54217”

Now Gauss did this at the age of nine. To be fair, he was the only child in the class that could do it. But we have to ask the question, “How many nine-year-old children could do that sum today?” Or even “How many of us could do that sum today?” And then we have to ask, “What has happened to make maths so difficult?”

Why not try this problem for yourself?

But here, should you want it, is Gauss’s probable working.

Firstly you need to find the total of the first 100 terms starting at 54217 and increasing by 17.

After 100 new numbers the figure would have increased by 1700, to give 55917. But you must only add 99 such numbers, so the hundredth term would be 55900. So the answer, still performed in Gauss’s head, would have been worked out like this:

54217 + 55900 = 110117

Why do we do this? – because as in the example above, we know that in a series of numbers the pairs all add up to the same as the two at the extremes—for 1 and 100 read 54217 and 55900.

Now you multiply this by 50, because we know the series is 100 numbers long, and half of that number of pairs. The quickest way is to multiply by 100 (just add 2 zeroes) and divide by 2.

110117 x 100 = 11011700
11011700 ÷ 2 = 55055850

However Gauss solved the question, this was an astounding mental feat for a nine-year-old, in any age. Thankfully, most of the maths we meet each day is a little simpler than that!