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How Long is a Piece of String?

The answers to the posers presented here can be found on the web, but what would be the fun in that? The real fun is in working these out from first principles.

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Recently, my chum Charles Pfeil set me to pondering over Palindromic Digital Clock Posers. Well, the little rascal just sent me an email containing a new question that had popped into his mind. In this communication, Charles spake as follows:
Mileage problem posed by Charles
Max, I was just looking at the dashboard in my car (see the attached photograph). With regards to distances travelled, I see two numbers. The first, 43,669, represents the total number of miles I’ve travelled thus far. The second, 3,658.4, represents the number of miles I’ve travelled since I last reset my trip meter, with the rightmost digit (the “.4”) representing tenths of a mile. I started to wonder what the best way would be to calculate at what mileage these two numbers would be identical if we were to assume that both 5-digit values represent integers (i.e., if we thought of the 3,658.4 value above as actually representing 36,584 for the purposes of comparing the two values).
Ooh! Well, if it’s a puzzle challenge Charles is looking for, I have three “old chestnuts” to share. Obviously, the answers to all of these can be found on the web, but what would be the fun in that? The real fun is in working these out from first principles. The funny thing is that it’s been so long since I thought of these that I’ve forgotten the solutions myself, so these will be posers for me as well as Charles — and also for you if you decide to take the challenge.  

Poser #1

The first poser involves two glasses we’ll call Glass A and Glass B, each containing a liquid of the same name. We also have a teaspoon that can hold a quantity of liquid we will call T. Both glasses contain the same volume of their respective liquids. I won’t tell you what this is, but – just to make all our lives easier – let’s state that it’s an integer multiple of T.
The two glasses problem (Click image to see a larger version).
So, here’s the poser. First, we take one teaspoon of liquid from Glass B, add it to the contents of Glass A, and stir Glass A so its contents are completely mixed. Next, we take one teaspoon of the mixture from Glass A, add it to the contents of Glass B, and stir Glass B so its contents are completely mixed. Now we repeat the process: one teaspoon from B to A and mix, then one teaspoon from A to B and mix. Finally, using mathematical reasoning or logical arguments (or a combination of both), explain which glass contains the highest concentration of its original liquid (if you see what I mean).  

Poser #2

Assume we have a metal cylinder in space, so we have zero gravity. Now assume that the cylinder is 100m in diameter, and its rotating around its central (long) axis at a rate of one revolution per minute. An astronaut is standing on the inside of the cylinder. He’s wearing magnetic boots, which aren’t shown here because I couldn’t be bothered to draw them. The astronaut is also wearing a space suit (also not shown here), which is fortunate for him because there’s no air in the cylinder.
The rotating cylinder problem (Click image to see a larger version).
The astronaut is 2m tall. He’s holding a tennis ball 1m “above” the “floor” of the cylinder. There are three questions: (a) What’s the name of the astronaut? (b) what’s the gravitational equivalent he’s experiencing due to centrifugal force? (c) If he lets go of the ball, where will it land on the surface of the cylinder (as measured from the mid-point between his feet)?  

Poser #3

This is another classic. The idea is that we have a “black box” containing a bunch of primitive logic gates. There are three inputs to the box (A, B, and C) and three outputs from the box (not_A, not_B, and not_C), where each output is the logical negation (inverse) of its corresponding input.
The three inversions with two NOT gates problem (Click image to see a larger version).
You can use as many AND and OR gates as you wish, and each of these gates can have as many inputs as you wish, but you can use only two NOT gates, and you can’t use any NAND, NOR, XOR, or XNOR gates.  

Bonus Question

How long is a piece of string?

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Charles Pfeil

Just have to say that I did remember (finally) that my question could be resolved with a little bit of simple Algebra. 43669+x = (3658.4+x)10
As for the posers offered by Max, I need to ponder them for a while.

David Ashton

Sorry, I’ll do the easy one first:

> How long is a piece of string?

A: About this colour (indicates with hands)…..

Charles Pfeil

Poser #1: Even if the liquids are mixed completely, starting with the second spoon the odds of the spoon containing the same ratio of molecules of each liquid as in the glass it is spooning from is infinitesimal. Can you say for the purpose of solving the poser, the ratio would indeed be exactly the same?

Charles Pfeil

How long is a piece of string? My biggest problem with this is each time I try to measure it, it changes or disappears. Therefore, I can only postulate that at any given moment it is twice as long as half its length.

Rick Curl

Hi Charles-
I think you may have gotten hold of some of the newfangled Quantum string.
At least that’s my theory.

-Rick

Charles Pfeil

Answer to Poser #1 = It depends on the relative size of the spoon to the glasses and what kind of round-off is allowed. However, the simplest case is when the glasses start with 1 only teaspoon of liquid in each of them and after the 4 exchanges, they have exactly the same amount of A and B in each of them. Both glasses have 1/2 teaspoon of A liquid and 1/2 teaspoon of B liquid. I have a graphic for this.

But, I believe if you start with 100 teaspoons in each glass, then depending on how much you round off the numbers, it won’t be exactly the same. I will give an example later alligator.

Crusty

Poser1 Could be interesting if the glass A contained an alkali and B contained an acid. These letters are arrivng slow on the screen when typing

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