## Poll

19 votes (46.34%) | |||

14 votes (34.14%) | |||

6 votes (14.63%) | |||

2 votes (4.87%) | |||

12 votes (29.26%) | |||

3 votes (7.31%) | |||

6 votes (14.63%) | |||

5 votes (12.19%) | |||

11 votes (26.82%) | |||

9 votes (21.95%) |

**41 members have voted**

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?

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?

“Extraordinary claims require extraordinary evidence.” -- Carl Sagan

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

Let x = sqrt(5*sqrt...

x^2 = 5*sqrt....

x^2/5 = x

x/5 = 1

x = 5

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.

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.

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.

I'll post a new problem shortly, but anybody else is welcome to do so before me.

“Extraordinary claims require extraordinary evidence.” -- Carl Sagan

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

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.

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?

March 28th, 2020 at 5:41:40 AM
permalink

Quote:djtehch34tDividing 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.

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.

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.

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

Find x.

“Extraordinary claims require extraordinary evidence.” -- Carl Sagan