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The story goes like this: A teacher, tired of managing a rowdy classroom, assigned the students to add up the first 100 positive integers. No calculators, just chalkboards. This was supposed to keep them busy for a good 15 to 20 minutes.
But one student wasnât interested in wasting time. Instead, he wrote the numbers in reverse order, lining them up under the original sequence. Suddenly, the problem became much simpler.
Adding vertically, he noticed a pattern: 1 + 100 makes 101, 2 + 99 makes 101, 3 + 98 makes 101, and so on. Every pair added up to 101, and there were 100 of these sums because the original list had 100 numbers.
Now, instead of adding each number individually, the student just multiplied: 100 Ă 101 = 10,100. But waitâheâd counted the whole sequence twice. Dividing by 2 gave the final answer: 5,050.
A problem meant to be a grind became a quick, elegant calculation. All it took was a fresh perspective and a little creativityâturning hard work into smart work.
#MathGenius #MathHacks #ThinkVertically
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Views : 41,490
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RYD date created : 2024-12-27T18:35:23.387296Z
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82 Comments
Top Comments of this video!! :3
For me, a slightly different method is more intuitive. 99+1=100, 98+2=100 etc. all the way to 51+49=100 we have 50 pairs adding up to 100 plus the middle 50. 50*100+50=5050
6 |
My math teacher was also a gardener. All his plants had square roots.
9 |
Alternative: add first and last (0+100), followed by (1+99), (2+98), etc. The result is 50X100 plus the 50 (centerpoint)= 5050.
11 |
With a uniform distribution like this, just take the statistical approach: The mean value is 50.5 â (100+1)/2 â and there are 100 values, so the answer is 50.5 x 100 = 5050.
12 |
This rule is how you find definite integrals with summation notation. It always fascinated me how they figured them out
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my favorite shortcut is squared numbers just equaling the sum of that many odd numbers sequentially. 2^2=1+3. 3^2=1+3+5, 7^2 =1+3+5+7+9+11+13, or easier said as "the first 7 odd numbers".
5 |
Mine is like this, find the exact middle number (value) then multiply by the number of digits; since the middle number (value) in 1 to 100 is 50.5, then it's 50.5*100 = 5050. If you want the middle to be an integer, then assume it's 1-99 first then add 100 to the result; the middle number is 50, 50*99 +100 = 5050
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I first heard this story from my math teach for jr highschool math, in the 80['s, It definitely inspired me to try to figure it out, and while I didn't figure it out in the time that he apparently did, i did figure it out on my own. I didn't even understand summation notion or anything like that yet. But the pattern makes sense when you think about it and try out a few smaller numbers first you can do by hand to confirm. But it took me closer to 20mins, not the few minuttes it took Gauss. I could have probably done the sums as quickly, but i was proud of myself for at least being able to get there eventually...
2 |
splitting the list in half. n/2 = 50 pairs
then we are adding 101, 50 times : (n+1)*(n/2)
with n = 100
(100 + 1)(100/2) = (101)(50) = 5050
equation to sum any number up to n
sum(n) = (n+1)*(n/2)
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A kĂ©plet ĂĄltalĂĄnos formĂĄja: (X(1)+X(n))Ă(n/2)
Ărtelmezve: A szĂĄmtani sorozat elemeinek összegĂ©t Ășgy kapjuk meg, hogy a szĂĄmtani sorozat elsĆ elemĂ©hez hozzĂĄadjuk a szĂĄmtani sorozat utolsĂł elemĂ©t, majd ezt megszorozzuk a szĂĄmtani sorozat elmeinek szĂĄmĂĄnak felĂ©vel. (ĂltalĂĄnos iskolai szĂĄmtan.)
1 |
One of my favorite weapons in MechWarrior and Battletech is the Gauss gun at (I think) 15 tons or so. Even though it's a fictional weapon, I'd still hate to get hit by any depleted uranium slug; that would hurt. A lot.
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Any complete list, i.e. 1-100 take the halfway point (50) multiply it by the last number (100) and add that halfway point number (5,050) Odd top number? (1-101) Take one more number past the closest even number halfway point (51) Multiply it by the top number (101) There's your answer (5,151)
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Gauss is still ahead of us some 200 years later...
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I thought it was because there were 50 combinations of 101...1+100 up to 50+ 51.
.
1 |
How TF did he even see that though?đ€·ââ
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Formula is n(n+1)/2
14 |
Gauss. If you have to guess, Gauss is the best guess
1 |
Gauss quantum computed the given supposedly consecutive list as parallel reversible computation.
Gauss was also the first to understand that imaginary numbers are really about parallel reversible inverse relation.
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I did the same thing in 6th grade. We had little chalkboards to use (no calculators). The teacher was surprised I got the answer so quickly.
6 |
@chrisk2557
1 week ago
That's really clever. I would have never thought to have done it by writing them vertically.
31 |