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:

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.

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 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.

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?

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.

Isn’t it funny how you can look at something and think it’s hard, then think about it a bit and realize it’s really simple (of course it works the other way too — you look at something and think it’s simple, then think about it a bit and realize it’s a lot harder than you first thought LOL)

Sorry, I’ll do the easy one first:

> How long is a piece of string?

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

We should perhaps note for non-native English speakers that the question “How long is a piece of string” is often used as a (colloquial, often humorous) response to a question such as “How long will it take?” or “How big is it?” when the length or size is unknown, infinite, or variable.

Baring this in mind, one of the best answers I heard was “Twice as long as from the middle to the end” (I’ve also heard, “twice as long as half its length”)

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?

“Can you say for the purpose of solving the poser, the ratio would indeed be exactly the same?” I can and I do!!!

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.

The way I always heard this said was, “twice as long as from the middle to the end” (see also my response to David Ashton above)

Hi Charles-

I think you may have gotten hold of some of the newfangled Quantum string.

At least that’s my theory.

-Rick

Oooh — that really is scraping the bottom of the barrel

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.

Just for giggles and grins, let’s assume that each glass starts off containing 1,000 teaspoons — but the actual number really doesn’t matter.

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

With regard to your slow letters — it could be that you are experiencing a local bubble in the quantum foam (see Is Time Truly an Illusion? https://www.clivemaxfield.com/is-time-truly-an-illusion/ )