Poll

9 votes (50%)
7 votes (38.88%)
3 votes (16.66%)
2 votes (11.11%)
6 votes (33.33%)
2 votes (11.11%)
3 votes (16.66%)
2 votes (11.11%)
8 votes (44.44%)
6 votes (33.33%)

18 members have voted

Wizard
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Wizard
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March 27th, 2020 at 7:35:48 PM permalink
It isn't easy finding math problems that I feel fit in the sweet spot of being hard enough to be beer worthy but not too hard that I couldn't solve it.

That said, this thread is for problems I feel are too easy for a beer, but might be a good challenge for the members who are not at the elite level here.

All are welcome to pose problems. Please put answers and solutions in spoiler tags until I've declared a winner.

That said, here is the first problem.

x = sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(5*sqrt(....))))))))))))))))

What is x?
It's not whether you win or lose; it's whether or not you had a good bet.
djtehch34t
djtehch34t
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March 27th, 2020 at 8:04:21 PM permalink
No beer club rule?


Let x = sqrt(5*sqrt...
x^2 = 5*sqrt....
x^2/5 = x
x/5 = 1
x = 5
Doc
Doc
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March 27th, 2020 at 8:05:06 PM permalink
x=5

The infinite (or semi-infinite) expression in the problem is equivalent to 5 raised to the power of an infinite series of terms, starting with 1/2 and each step involving adding 1 and dividing by 2:
(((...((((((((1/2)+1)/2)+1)/2)+1)/2+ ...)

The sum approaches 1. For various finite sub-series of the terms, the exponents of 5 would be 1/2, 3/4, 7/8, 15/16, 31/32, 63/64, etc.

x = 5^1 =5.
Wizard
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Wizard
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March 27th, 2020 at 8:24:42 PM permalink
I agree, the answer is as quoted above.

I'll post a new problem shortly, but anybody else is welcome to do so before me.
It's not whether you win or lose; it's whether or not you had a good bet.
ssho88
ssho88
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March 27th, 2020 at 8:49:07 PM permalink

Anaswer = 5^0.5 * 5^0.25 * 5^0.125 * 5^0.0625 *........
= 5^(0.5+0.25+0.125+0.0625+ .....)
= 5^1
= 5

0.5+0.25+0.125+0.0625+ .... is a geometric series, r=0.5, Sn = 0.5( 1- 0.5^infinity)/(1-0.5) = 2 * 0.5 = 1
ChesterDog
ChesterDog 
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March 27th, 2020 at 9:46:00 PM permalink
Quote: djtehch34t


Let x = sqrt(5*sqrt...
x^2 = 5*sqrt....
x^2/5 = x
...


Instead of next dividing both sides by x, try subtracting x from both sides and factoring.
djtehch34t
djtehch34t
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March 28th, 2020 at 4:56:46 AM permalink
Quote: ChesterDog


Instead of next dividing both sides by x, try subtracting x from both sides and factoring.




Dividing by x should be legit because we can clearly tell that x > 0?
ChesterDog
ChesterDog 
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March 28th, 2020 at 5:41:40 AM permalink

Quote: djtehch34t

Dividing by x should be legit because we can clearly tell that x > 0?



It took me a while, but now it's clear to me that x must be greater than zero.

Thanks.
charliepatrick
charliepatrick
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March 28th, 2020 at 4:07:01 PM permalink
If x = SQRT(5*SQRT(5*....)) = SQRT(5*x). It then seems obvious x=5, but mathematically squaring each side to get x^2 = x*5, so x=5.
Wizard
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Wizard
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March 29th, 2020 at 8:09:01 PM permalink
Here is your next puzzle that is too simple to be beer-worthy, but is nevertheless pretty challenging.



A 3-4-5 triangle is inscribed in a square of length x.

Find x.
It's not whether you win or lose; it's whether or not you had a good bet.

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