Hey, this is Presh Talwalkar and you're watching mind your decisions
Here's a very fun number puzzle
You start out with three identical copies of each number, and you need to make it equal to six
The puzzle is to make each equation true, using common mathematical operations
There are two rules
First, you cannot introduce any new digits
So for example, the operation of cube root would not be allowed, because it involves the digit 3
The second rule is you must have the result equal to 6. You cannot use the not equal symbol.
I thank Tyler Cenko for first
Suggesting me this problem, and I thank Caio Cerqueira from Brazil for also suggesting it.
I also want to present a note, my videos will make you so smart that even difficult problems like this will become easy to you
In videos like this you'll see people
Asking for harder problems and it's the best proof that these videos are teaching math to people around the world
So, can you figure it out?
Give this puzzle a try, and when you're ready, keep watching the video for a solution
So, how can we solve this puzzle
I will mention that some of the equations have multiple answers
Due to time constraints, I'm only going to present some of the solutions and you may have found another
creative method to solve one of the equations
Let everyone know your answer in the comments
We'll start solving these by going from some of the more straightforward and easy ones, to then some of the more involved in harder problems
Let's start out with 2 2 2 to equals 6
We can solve this by using the equation 2 plus 2 plus 2 is equal to 6
So we've already figured out 1
Now 6 6 6 is also similar we can take 6 plus 6 minus 6 to get to 6
Or we can also do 6 times 6 over 6 is equal to 6
For our aggregation of solutions. Let's just go with the first equation
Now let's solve 7 7 7 equals 6 we can do this as 7 minus 7 over 7 is equal to 6
We get a very similar solution for 555 we can take 5 plus 5 over 5 is equal to 6
So we're making good progress
Now, let's solve 3 3 3 is equal to 6
One way to solve this is 3 times 3 minus 3 is equal to 6
But there are actually several more solutions for this
Another one is 3 factorial plus 3 minus 3 is equal to 6
We can also do 3 factorial times 3 over 3 is equal to 6
Or we can do something like the square root of the quantity
3 times 3 and then we can add 3 and that's also equal to 6
So for presentation purposes, I'll just take 3 times 3 minus 3 is equal to 6
Now let's solve 9 9 9 is equal to 6
Well one trick we can use is let's take the square root of each of these 9s
Here's a handy mathematicians trick.
Since the square root of 9 is equal to 3 we have reduced the problem to solving 3 3 3 is equal to 6
And this problem has previously been solved
So any of our solutions for 3 3 3 is equal to 6 will also apply
To this 9 9 9 once we've applied the square root to each of the 9s
So we could solve this as the square root of 9 times the square root of 9 minus the square root of 9
That's equal to 6 or we take the quantity the square root of 9 factorial
Plus the square root of 9 minus the square root of 9 is equal to 6 or we have some other solutions
Like the square root of 9
factorial times 9 over 9 is equal to 6 or
the quantity
square root of 9 times the square root of 9 divided by
The square root of nine we take the whole thing and the factorial of that and that is also equal to six
For presentation purposes. I'll just take this first equation
Now let's go to four for four is equal to six well once again we can take the square root of each of these numbers
again, we have our handy trick since the square root of four is equal 2 we have reduced the problem to 2 2 2 is equal to 6
And that's previously been solved.
For example we can take the sum of these numbers and that'll be equal to 6
But if you don't want to do that, you can also do the quantity 4 minus 4 over 4
And we take the factorial of that quantity
We can also do the square root of 4 plus 4 over 4
we take that entire quantity and then take the factorial of that and that's also equal to 6, so
for presentation purposes, let's just take this first equation.
Now let's solve 10 10 10 is equal to 6
Let's do this in steps
We'll do 10 minus 10 over 10, and at this point our result is equal to 9
Well now take the square root of this to get to a result of 3
Then we will take the factorial
So that's how we can get 10 10 10 - equals 6
Now, what about 1 1 1?
well notice if we add the 3 numbers we get to 3 and then we'll take the factorial and that's equal to 6 so
Now, how do we solve 0 0 0 is equal to 6? Well first we'll take the factorial of each of these numbers
We again use a mathematician's trick
Since 0 factorial is equal to 1 we have reduced the problem to 1 1 1 is equal to 6, which has previously been solved.
So we add up these numbers and then we take the factorial of the sum and that will be equal to 6
Finally how do we solve 8 8 8 is equal to 6? Many people consider this to be the hardest of all of these equations
So we'll work in steps
We'll take 8 plus 8 and that's equal to 16
We'll take the square root of that and the square root of 16 is equal to 4
We then take the square root of the square root of that
So the square root of the square root of 16 will be equal to 2
now we take 8 minus 2 and that's equal to 6
so that's one way to solve this
But another way that you could solve this is you could take 8 plus 8 over 8, and that'll be equal to 9
We then take the square root of 9, which will be equal to 3 and then we take the factorial of 3
and that's equal to 6
So that's it. We figured out how to get 6 in each of these equations
Did you figure it out?
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