I came across this problem, can you count up all the numbers from 1 to 100?
Without actually counting them like a sucker. Also without doing the calculations on a computer
like basement dweller. Can you think of another way to do it really
fast? I couldn’t do it. So I just looked it up.
Turns out there’s a story behind it and it goes like this:
In 1784 there was this 7 year old kid, Carl Gauss. He was in class and their teacher gave
them this task: count up the numbers from one to a hundred.
Not as a real lesson mind you, he just wanted to keep them busy for a bit so he could work
on his novel. I guess it’s kind of like today when your
teacher puts on Planet Earth in science class Or Air Bud in phys ed.
So everyone quietly gets to work. But Carl Gauss was like: “no bitch”. And
he came up with an easier way to do it. The key to figuring it out, was noticing if
you have a sequence of numbers like this and you add the first and the last numbers, they
add up to the same thing as adding up the second number and second to last numbers.
And the same thing goes for adding up the third and third to last.
This number is one bigger than the one we just did, this number 1 smaller than the one
we just did. So if we add them together, it’s going to be one bigger and one smaller than
the sum we just did. So we’re going to get the same number.
And that works all the way to the middle. We can basically just fold the number line
in half and all the pairs are going to add up to the same number.
Playing with the numbers like this is the secret to adding them up really fast.
Because it lets us multiply this number, the value of the pairs, by the number of pairs.
And it’s exactly the same as if we had added them up one by one. We’re just sort of adding
them up in a different order. So for our equation.
We want to take the first number. Add the last number to get the value of the pairs.
Then we just have to multiply that by the number of pairs.
So you know, 1, 2, 3, 4… Counting is for Philistines!
And what if there’s too many numbers to count!
You can’t count em! There’s too many!
So let’s find a better way to do it. Let’s say we wanted to count the number
of numbers from 3 to 6. And we know the answer’s 4. So let’s see
if we can make 4 from the number’s we’re working from, 3 and 6.
Could start with subtracting them. But if we subtract them we get 1 away.
If we were counting the number of numbers from 7 to 15.
There’s 9 numbers. But if we subtract them we also get 1 away.
Because subtraction is giving us the difference between the two numbers.
Whereas because of the specific way we worded the problem, we want to “count from something
to something”. We want to take count of that first number
that gets subtracted. So to make up for the number that gets lost,
we just have to add 1. And that’s it really.
Take the last number, subtract the first number to get the difference between them.
Add 1 to make up for that 1 that gets lost in the subtraction.
Divide by 2, to go from having the number of numbers, to having the number of pairs.
And then multiply that by the value of the pairs.
So if we want to add up all the numbers from one to a hundred. We go shuh ba doop. Mrum.
Hwweeoo. Wshhoo. Wackaw the answer. And we can check that with a computer.
I guess computers are OK, I’m sorry for calling you basement dwellers.
And so he came up with the answer really fast, and the teacher was like “whaaat?!”.
He just, he just couldn’t believe it. You know, kind of like that scene in a movie.
“I’m done” “What? Oh. You know you don’t have to
tell me when you finished a portion of it. You brown nosing little sh…”
“No, I’m done the whole thing” “Whhaat?!”
“Is it even possible? DO they have enough time to get this done. Mreh, I’m so mysterious”
“No no no, it’s a test in problem solving, I want to see how they approach the task because
that’s really the key in what we’re looking fo…”
“I am no man” “… What?”
“I mean I’m done” “What?! I guess it just goes to show, when
sometime… sometimes when you, uh, don’t expect anything of so…, sometimes the people
you don’t expect…” And that’s the equation and we can use it
on really really long sequences if we wanted. And we can check that.
The equation he came up with was actually easier.
Because when you’re counting from 1 to something you can kind of skip all this nonsense. There’s
4 numbers counting from 1 to 4. There’s 100 numbers when you’re counting from 1
to 100. So we don’t have to calculate the number
of numbers, that last number is just the number of numbers.
Also the “first number” when counting from 1 to something, is just going to be 1.
And then if we’ve only got the 1 number we’re counting to, then let’s just call
“the number”. So you’ve got the number of pairs, and the
value of each pair. And you’ll get the same number adding up
to 100.
What I love about it is that this feels like a silly observation, something you might have
noticed just playing around looking at stuff in the room.
But if you think to do so, you can also use it to solve a problem.
Speaking of mathematics, This episode is brought to you by: piiiiii
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