Can You Solve The Four 9s Puzzle?

0:00 puzzle
1:42 brainstorming
3:31 main solution
4:16 simple
5:59 trig
9:44 clever
10:19 genius

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[DrFunkman]

Damn, I spent around 15 minutes looking at the thumbnail and trying to brute force the answer, not knowing I was allowed to just use 99

[wyattstevens]

99+9/9 (who said there had to be an operation between *every* pair of 9s?)

[therat]

I got to 118.62... Close enough for engineering

[HanzoHimemiya]

basic solutions: 99+9/9
adult solutions: logaritms, trigonometric an exponents

[SanketAlekar]

Anything that uses "0.9" or "99%" isn't "pure", as it uses 10 implicitly.

[YellowBunny]

In my opinion sqrt() shouldn't be allowed as it contains a 2 that we just don't write by convention and % should definitely not be allowed because it is just a symbol for division by 100.

Edit: Concatenation and stuff like .9 also feels like it's just cheating.

[victorkaplansky]

The last solution is known universal solution which works not only for 9s and 100 as result.

[leickrobinson]

6:43 Small nitpick, but you say arccos of “zero degrees”, but the input to arccos is not an angle. It should be just arccos of zero.

[TheJaguar]

That last one is just insane. I was looking for a log-based solution, but that is nuts.

[pranavyadav]

9 || 9 + 9 ÷ 9 = 100

|| is the sign for concatenation. Saw it on Numberphile's video, Matt Parker.

[cmilkau]

I like the cheekyness of the last solution as you can use it to make any number from almost any 4 copies of the same digit (only 0 and 1 fail) just by adjusting the number of square roots

[joachimsimon]

Basically we can use 3 and 6 for the 9 since
• 3 = sqrt 9
• 6 = 3! = (sqrt 9)!
Furthermore we can also use 1/2 because 1/2 = cos (tan-1(sqrt 3))

Finally we have
(1/2)^-6 + 6*6 = 100

[darryllmaybe]

That last one is actually just so beautiful and can be applied to literally any number so long as they are all the same number since it derives its answer from the properties of logs rather tricks with the specific number.

Technically, "(x/.x)(x/.x) will also evaluate to 100 as well regardless of x's value, but that's a little cheap since ".x" implies dividing by 10 or using a 0, plus it's not nearly as elegant.

[MastuurOogwaay]

I believe i found a really good solution that doesnt 'twist' any of the rules: (9+cos(9!)) * (9+cos(9!)) = 100 because cos(9!) = 1


Or another one: arccos(9-9) + 9 + cos(9!) = 100

[orionspur]

Gluing digits together, or using the decimal point, or using the % sign, are all quietly invoking 10s. Seems unfair.

[Sam_]

I have an interesting solution

9.9with a line on top of nine multiply

[navadeepkandre]

I am watching you, after a lot of time... You're still the same amazing, when I used to watch you earlier... Now I think I will continue watching your content.

[alexanderschreiter]

That last solution is beautiful as it’s a wonderful general case solution allowing you to replace the nice with any number as long as they are all the same as their purpose is only to cancel the logs and you can get any integer you want by just increasing or decreasing the number of sqrt in your numerator quantity

[tlbqgdv]

"What an incredible solution" and What an incredible solution (the log example) ... RC: and it's goodnight from me, RB: and it's good night from him.

[hhhhhh]

i remember numberphile showing off the last solution years ago for the four 4s problem, glad to see it's still around
when you said you just wanna see 100 displayed in the calculator in the intro, my immediate thought was to just keep pressing factorial and square root until it happens to get close enough to 100 to round to it in whatever floating point data type the calculator is using