## Poll

21 votes (45.65%) | |||

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3 votes (6.52%) | |||

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5 votes (10.86%) | |||

12 votes (26.08%) | |||

10 votes (21.73%) |

**46 members have voted**

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?

Let x = sqrt(5*sqrt...

x^2 = 5*sqrt....

x^2/5 = x

x/5 = 1

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.

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

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

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.

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?

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.

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

Find x.

Quote:Wizard

(snip)

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

Find x.

Could I possibly find an answer or approximate answer if I had:

***:

^^^:

Graph paper would certainly be useful to get an estimate, but for full credit, I want to see an algebraic solution.

See image : https://imge.to/i/yV3aUG

From blue triangle, cos(θ) = x/4,

and

From red triangle, cos(θ) = a/3

Therefore, a/3= x/4, a=3x/4, and b = x/4

b^2+x^2 = 4^2

(x/4)^2 + x^2 = 16

x= 16/(17)^0.5 = 3.8806

Note: I know that this formula works for at least one type of "area of a triangle inside a square " problem, but I don't know if it works for the type you posted.

Length of side "X" = Square root of the "area of the square" =

Square root of ("Area of the Triangle" x 2)

Using the above, for a triangle where: 3 = base and 4 = height , then X would be = 3.46... (or square root of 12 to be exact).

X = √ 12 (where X must be a positive number )

I hope this is correct, as I will not be attempting any more answers to this problem if it is wrong (in other words, "too hard for me")

Edit:

Going by the answer above by ssho88, I am going to say my answer is almost certainly wrong.

Nice puzzle - thanks.Quote:WizardHere 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.

Thus if the base of blue is x, the base of red is 3/4 x. So the side of blue is 1/4x.

(Academically you can now work your way round to show the all parts clockwise are x 1/4x 3/4x 3/16x 13/16x x.)

Looking at blue triangle x^2 + (1/4 x)^2 = 16 = 17/16 x^2. So x^2 = 256/17.

If you do the same maths with the others you get the same result (e.g. 25 = (169+256)/256 x^2 = 425/256 x^2).

X = SQRT (256/17).

Quote:gordonm888Let's hope there is not a pandemic lockdown during the next total eclipse in 2024.

Amen to that.

The state of Denial has a population of 20 million. They are in the early states of an epidemic. They estimate one person in 500 will need to be on a ventilator when the epidemic is at its peak. How many ventilators should they have on hand to have a 95% chance of having enough?

Quote:ssho8839671 ?

mean = 1/500 = 0.002, Var = 0.002(assumed), SD =0.04472

For > 95% of chance, Z > -1.645

(X -40000)/0.04472/(20,000,000)^0.5 > -1.645

X > 39671

I am not sure about the Variance value . . .

Average = Np = 40k.

SD = SQRT(N p q) = SQRT(20m / 500 * 499/500 ) = 199.7999. (Since q is nearly 1, this happens to be SQRT (Average) )

Number of SDs for 90% (i.e. 5% at either end) = 1.644854 ( https://en.wikipedia.org/wiki/Standard_deviation ).

So number (over average) needed is 328.642.

Hence number to ensure 95% = 40k + 328.642 = 40329.

Quote:charliepatrickI think ssho88 has the figure the wrong way up (so to speak) but agree with his idea.

If the government is going to buy the normal amount it might as well buy a few extra. The main problem is guessing whether 1 in 500 is correct in the first place. For instance if it was 1 in 496 then you'd now only have a 50% chance of having enough.N=20m, p=1/500, q=499/500.

Average = Np = 40k.

SD = SQRT(N p q) = SQRT(20m / 500 * 499/500 ) = 199.7999. (Since q is nearly 1, this happens to be SQRT (Average) )

Number of SDs for 90% (i.e. 5% at either end) = 1.644854 ( https://en.wikipedia.org/wiki/Standard_deviation ).

So number (over average) needed is 328.642.

Hence number to ensure 95% = 40k + 328.642 = 40329.

I think we have the same mean and variance, mean and variance calculated by me is 40000 and 200 [0.04472 * (20,000,000)^0.5]

(X - 40000)/200 >-1.645

X > 39671

The difference is I used Z > -1.645 instead of +1.645 ??? Which one is correct ? I am not sure.

Quote:ssho88Quote:charliepatrickI think ssho88 has the figure the wrong way up (so to speak) but agree with his idea.

If the government is going to buy the normal amount it might as well buy a few extra. The main problem is guessing whether 1 in 500 is correct in the first place. For instance if it was 1 in 496 then you'd now only have a 50% chance of having enough.N=20m, p=1/500, q=499/500.

Average = Np = 40k.

SD = SQRT(N p q) = SQRT(20m / 500 * 499/500 ) = 199.7999. (Since q is nearly 1, this happens to be SQRT (Average) )

Number of SDs for 90% (i.e. 5% at either end) = 1.644854 ( https://en.wikipedia.org/wiki/Standard_deviation ).

So number (over average) needed is 328.642.

Hence number to ensure 95% = 40k + 328.642 = 40329.

I think we have the same mean and variance, mean and variance calculated by me is 40000 and 200 [0.04472 * (20,000,000)^0.5]

(X - 40000)/200 >-1.645

X > 39671

The difference is I used Z > -1.645 instead of +1.645 ??? Which one is correct ? I am not sure.

Presumably, to be 95% confident, you find the value such that 90% of the values are within that many SDs on either side of the mean; 5% will be greater than the mean + that many SDs, so 95% are less than that value.

Quote:ThatDonGuyQuote:ssho88Quote:charliepatrickI think ssho88 has the figure the wrong way up (so to speak) but agree with his idea.

If the government is going to buy the normal amount it might as well buy a few extra. The main problem is guessing whether 1 in 500 is correct in the first place. For instance if it was 1 in 496 then you'd now only have a 50% chance of having enough.N=20m, p=1/500, q=499/500.

Average = Np = 40k.

SD = SQRT(N p q) = SQRT(20m / 500 * 499/500 ) = 199.7999. (Since q is nearly 1, this happens to be SQRT (Average) )

Number of SDs for 90% (i.e. 5% at either end) = 1.644854 ( https://en.wikipedia.org/wiki/Standard_deviation ).

So number (over average) needed is 328.642.

Hence number to ensure 95% = 40k + 328.642 = 40329.

I think we have the same mean and variance, mean and variance calculated by me is 40000 and 200 [0.04472 * (20,000,000)^0.5]

(X - 40000)/200 >-1.645

X > 39671

The difference is I used Z > -1.645 instead of +1.645 ??? Which one is correct ? I am not sure.

Presumably, to be 95% confident, you find the value such that 90% of the values are within that many SDs on either side of the mean; 5% will be greater than the mean + that many SDs, so 95% are less than that value.

I agree. "to have a 95% chance of having enough" means LESS THAN that Z value. I interpreted it as MORE THAN in previous calculations.

The problem meant to ask how many ventilators should they have so that the probability of running out would be 5%, which is equivalent to a 95% chance of not running out.

Quote:ChumpChangeHey 19

Quote:WizardHere is the next one.

edit

For anyone who wants to win a beer or sugar free rockstar from this question, what shoe is featured in the picture?

Quote:ssho885 +5 x 2 = 15

Hint:

Show your work:

ss + ss + ss = 30, ss = 10, s = 3.16227766...

m + m + ss = 20, m =5

ll + ll + m = 13, ll = 4, l = 2

now look very closely at the picture, preferably on something other than your phone

s + mllss x l

3.16227766 + (5 x 4 x 10) x 2 = 403.16227766

Quote:EdCollinsAh, nevermind. Just now noticed the man is different.

Oh yeah. Heh. TomG is right about cell phones.

Quote:TomGHint:

there is a place they try to trick you

Show your work:Let shoe = s, man = m, lifting strap = l

ss + ss + ss = 30, ss = 10, s = 3.16227766...

m + m + ss = 20, m =5

ll + ll + m = 13, ll = 4, l = 2

now look very closely at the picture, preferably on something other than your phone

s + mllss x l

3.16227766 + (5 x 4 x 10) x 2 = 403.16227766

yeah, LOL

Quote:TomGHint:

there is a place they try to trick you

Show your work:Let shoe = s, man = m, lifting strap = l

ss + ss + ss = 30, ss = 10, s = 3.16227766...

m + m + ss = 20, m =5

ll + ll + m = 13, ll = 4, l = 2

now look very closely at the picture, preferably on something other than your phone

s + mllss x l

3.16227766 + (5 x 4 x 10) x 2 = 403.16227766

I have a different interpretation.

I interpret the two shoes to mean s + s. We do agree that two ties are 2t, so why would two shoes not be 2s?

Using my logic, the answer is 43.

With 10% of the population having COVID-19, the world is in extreme crisis. Thankfully, a new Pharma company has just created an instant COVID-19 test that is 90% accurate. World leaders have mandated that everyone take the test in the hopes this will enable the virus to finally be contained. You dutifully show up at Walmart (wearing your N95 mask and staying six feet away from everyone else) to get tested in the pharmacy. Bad news: the test is positive for COVID-19.

What is the probability you actually have COVID-19?

Quote:WizardQuote:TomGHint:

there is a place they try to trick you

Show your work:Let shoe = s, man = m, lifting strap = l

ss + ss + ss = 30, ss = 10, s = 3.16227766...

m + m + ss = 20, m =5

ll + ll + m = 13, ll = 4, l = 2

now look very closely at the picture, preferably on something other than your phone

s + mllss x l

3.16227766 + (5 x 4 x 10) x 2 = 403.16227766

I have a different interpretation.

I interpret the two shoes to mean s + s. We do agree that two ties are 2t, so why would two shoes not be 2s?

Using my logic, the answer is 43.

Quote:unJonWhat makes you think Tom is saying two ties is 2t and not t^2? It’s the same answer since t=2

Good point. However, I'm sticking with my answer, but admit it is subject to interpretation.

Quote:unJonThis one will be truly easy for those on this forum, but it’s an old classic and I used to use it as an interview question (updated for the current crisis):

With 10% of the population having COVID-19, the world is in extreme crisis. Thankfully, a new Pharma company has just created an instant COVID-19 test that is 90% accurate. World leaders have mandated that everyone take the test in the hopes this will enable the virus to finally be contained. You dutifully show up at Walmart (wearing your N95 mask and staying six feet away from everyone else) to get tested in the pharmacy. Bad news: the test is positive for COVID-19.

What is the probability you actually have COVID-19?

This is a classic problem in conditional probability.

...I keep forgetting to count the number of items - for example, where there are two shoes in the first equation and just one in the last one

Quote:WizardThis is a classic problem in conditional probability.

I never liked these either, because you have to assume that the condition for which you are testing is uniform - in this case, the probability that you have COVID-19 is 10%.

Suppose that you could determine that a shoe represents the number 5. Do two shoes right beside each other represent 2*5 = 10, or 5*5 = 25, or a double-digit-symbol = 55? With that much ambiguity in the symbols -- even without the hidden shoes and straps added to the man (which I didn't see) -- I just thought the numerous ways to reverse-interpret a set of equations was not something I wanted to try.

Quote:DocI punted on the symbols problem from the very beginning, because I thought there were just too many ways to interpret the combinations of symbols.

Suppose that you could determine that a shoe represents the number 5. Do two shoes right beside each other represent 2*5 = 10, or 5*5 = 25, or a double-digit-symbol = 55? With that much ambiguity in the symbols -- even without the hidden shoes and straps added to the man (which I didn't see) -- I just thought the numerous ways to reverse-interpret a set of equations was not something I wanted to try.

Almost always, two symbols = 2 x one symbol in these.

LOL, I went from the first puzzle straight to the last post.

A rectangle bisects a cylinder, going through the center of the bottom and top. The perimeter of the rectangle is 6. What is the maximum volume of the cylinder?

Note: I had "two answers" and this one seemed the most likely to me

Volume = π ≈ 3.14

Edit (about 3 pm):

I think the answer by the Joeman below is lot better than mine, because:

1. He found the answer more efficiently than me (I used a "trial and error" type of way),

2. He can prove that his answer is correct with a formula/method.