# 538 Riddler Express

Posted this in whatever popped up in search, but it was in “professional” section, so, new thread!

For this week’s (4/29/2022) numerical puzzler:
I sometimes read these to see how to solve them, but don’t normally take time out of my busy posting schedule.

## Riddler Express

There are many fractions with a numerator of 1 whose decimal expansions don’t go on to infinitely many decimal places. For example, 1/4 is equivalent to the decimal 0.25, and 1/500 is equivalent to 0.002. However, the decimal expansion of 1/3 is 0.33333 …, a decimal that never terminates.

If you were to add up all these numbers — fractions with a numerator of 1 whose decimal expansions don’t go on forever — what would be the sum? (Note: Before you ask, let’s include the fraction 1/1 in this group.)

I thought this was pretty easy.

2.5

my reckonin’

Only numbers with denominators that are factors of ONLY 2 and 5 qualify, cuz, Base 10.
Solution = Sum (i=0 to infinity) {Sum (j=0 to infinity) (1/5^i * 1/2^j)}
(If someone wants to mathify that equation, that would be awesome.)

Each j Sum is 1/5^i * 2
So, factor the 2 to the front, and you get:
2 * Sum (i=0 to infinity) (1/5^i )
= 2 * 1/(1-(1/5)) = 2 * 5/4 = 2.5

Somebody please check my math, as I could forget to carry the 1, mghey.
Of course, the answer could also be 0, since there is a negative fraction for each positive fraction.

Typo

I get the same answer as the OP.

Not sure how that happened. I pasted, and it’s correct on 538. Unless, it was incorrect earlier on 538. I’ll fix mine.

Hey! I got it right!

May 6, 2022:

You are on the 10th floor of a tower and want to exit on the first floor. You get into the elevator and hit 1. However, this elevator is malfunctioning in a specific way. When you hit 1, it correctly registers the request to descend, but it randomly selects some floor below your current floor (including the first floor). The car then stops at that floor. If it’s not the first floor, you again hit 1 and the process repeats.

Assuming you are the only passenger on the elevator, how many floors on average will it stop at (including your final stop, the first floor) until you exit?

Hmm, seems like way too many paths to floor 1.

Maybe determine the avg number of floors for each starting floor, starting at Floor 2, see if there is a pattern?
So,
Start on Floor 2. Avg = 1.
Start on Floor 3: Avg = 1/2 * 1 + 1/2 * 2 = 1.5
Hmm, that’s 1/2 more than Floor 2.
Start on Floor 4: Avg =
1/3 * 1 (land on Floor 1) +
1/3 * (1+ 1) (land on Floor 2) +
1/3 * (1+ 1.5) (land on Floor 3) = 1/3 +2/3 + 5/6 = 1 and 5/6 (or 11/6, or 22/12).
Hmm, that’s 1/3 more than Floor 3
Start on Floor 5: Avg =
1/4 * 1 (land on Floor 1) +
1/4 * (1+ 1) (land on Floor 2) +
1/4 * (1+ 1.5) (land on Floor 3) +
1/4 * (1+ 11/6) (land on Floor 4)
= 1/4 +3/4 + 3/8 + 17/24 = 25 /12,

Hmm, that’s 3/12 (1/4) more than Floor 4 (22/12).

So, 1 + 1/2 + 1/3 + … + 1/9 = 2.82896…

Last week’s was pretty easy. I think I posted elsewhere. And they Jeopardy’ed the question by adding an additional clue to make it extremely easy, like Jeopardy.

This week is a bit different.

Summary

There are eight paths, each equally possible.
There are only four ending points possible:

1. Three radially: Distance = 4 inches. One Path.
2. Two radially, one tangential: Distance = (3^2 + 1^2)^0.5 = 10^0.5. Three Paths.
3. One radially, two tangential: Distance = (2^2 + 2^2)^0.5 = 8^0.5. Three Paths.
4. Zero radially, three tangential: Distance = (1^2 + 3^2)^0.5 = 10^0.5. One Path.

So 0.125 * { (1 * 4) + (3 * 10^0.5) + (3 * 8^0.5) + (1 * 10^0.5) }
Whatever that turns out to be.

Need to divide by the total number of paths.

Oh, let me fix that.

I think this is wrong.

Consider the possibilities of Radial (Given), Tangential, Radial, Tangential. This would be in your group 3. Where are you, assuming you start at (0,0) on a cartesian grid?

A radial after a Tangential moves you directly away from the center, NOT perpendicular to the tangential

So I think you start at:
0,0
0,1 (first move)
1,1 (Second move)
1.707107, 1.707107 (3rd move)
somewhere 2.631126 from (0,0) after the 4th move

Oh, I think you’re right. That seems a lot harder than “express” would imply.

Also, are we to assume that moving tangentially right or left, once tangentially is chosen, are of equal probability? (Or, does it make a difference?)

Making more drawings…

After much thought, I think it does NOT make a difference.

After reading more, I think Transition Left followed by Transition Right is not allowed:

(To be clear, when Amare moves tangentially, he is still moving farther away from the center of the web, at least a little bit.)

So, can’t move closer to the web. However, after a radial move, the transitional move, left or right, will be farther and WLOG, no diff.

After think about it more, I think Transition Left followed by Transition Right makes no difference to the distance to the original starting position.

Seems to me that if I am at distance x, then I either get to distance x+1 if moving radially, or (x^2+1)^0.5 if move tangentially. Can start with x = 1, and then enumerate the 8 cases, and take an average. I get ~2.95.

{Edited to spoil}

On the menu bar, there is a cog. When pressed, the cog shows a few “tools,” one of which is a “Hide Spoilers” tool.

Thanks!

I concur, but to more decimal points I get 2.9509032

Bump, for today’s Riddler Express:

What interest rate is exactly doubled when using the Rule of 72?

Spoilered:

It’s some irrational number, since the interest rate is both inside a Ln() and outside an Ln(). I have, so far, using Excel: 7.846871453015% and that is “exact” only to 14 decimals. Maybe there is some newer math I’m not familiar with.

Pretty sure you can’t get a closed-form result. Approximation is the best you can do.

This weeks Riddler Classic is a Macauley Duration calculation, No?