We come from the future
We come from the future

# Here Are Three Variations On A Classic Puzzle – Can You Solve Them All?

I've always enjoyed water-distribution puzzles. They are simple in structure and usually straightforwardly posed, but their plain presentation belies their challenging nature. Here are three classic variations on the theme to wrap your head around.

## Sunday Puzzle(s) #15: Pouring And Partitioning Water

I. You are handed two water glasses. The smaller glass can hold exactly four ounces of water, the larger exactly nine ounces. With nothing more than these two glasses and an endless supply of water, your task is to measure exactly six ounces of water. You can fill or empty either glass as many times as you wish. In the interest of conserving water, try to do this in as few steps as possible. What's the best you can do?

II. You return to the unlimited water supply. This time, you have in your possession two glasses measuring seven and eleven ounces. Again, you are permitted to fill or empty either glass. How many steps are required to fill one of the containers with exactly six ounces of water?

III. Given a five ounce glass and a three ounce glass, measure exactly four ounces. Again, your water supply is limitless and you can fill and empty glasses as you please.

We'll be back next week with the solutions – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. (Be sure to include "Sunday Puzzle" in the subject line.)

Art by Jim Cooke

## SOLUTION To Sunday Puzzle #14: How Much Of Yourself Can You See In A Mirror?

As some of you pointed out in the comments, this is as much a physics problem as it is a puzzle, and the solution involves a basic understanding of optics. The answer: You need a mirror at least half your height to see the entire length of your body. Said another way: It is possible to see twice as much of yourself as the length of any normal wall-mounted mirror.

As many of you pointed out, such a setup does not technically allow you to see YOUR ENTIRE BODY. The rear half of your person – not to mention your shoes – are not visible. While technically correct, this was not the answer I was looking for when I posed this puzzle. That being said, I will cede that "technically" correct is my favorite kind of correct, so I'm happy to award 5,000 arbitrary Internet points to the pedants, on this one.

Anyway. For the half-as-tall solution to work, a mirror half your height must be hung on a wall such that the top of the mirror is level with the halfway point between the top of your head and your eyes. The reasoning for this can be distilled down to two points. Number one: To see any point on your body, light from that body part must bounce off the mirror and back to your eye. Number two: When light reflects off a smooth surface (such as that of a standard mirror), it does so at an angle equal to the angle at which it hit the surface. Behold, my formidable artistic abilities:

(I thought I could sketch this out myself. In retrospect, I probably should have had our art department draw it up, but this should do in a pinch.)

Note that when the mirror is hung as described above, it is possible to see your feet reflected back at you from the bottommost portion of the mirror. Note, also, that the amount of your body that is visible in the mirror remains constant, regardless of your proximity to the mirror. Moving farther away from the mirror will change the angle at which light arrives at and leaves the mirror (note that angles A and B are equal, but significantly smaller than the analogous angles C and D), but will not change the amount of yourself that is visible.

Some of you may find it useful to think of the mirror as a window through which one peers at a non-superimposable, mirrored version of yourself. In the diagram above, the stick figure on the far left represents the "virtual image" that is visible to the stick figure in the middle. Because the mirror is perfectly flat, the virtual image appears to be the same size as the middle stick figure. Similarly, the virtual stick figure appears to be the same distance behind the mirror as the real, middle stick figure is in front of the mirror.

While potentially helpful, this line of thinking can be confusing. I suspect it is why, conceptually, a lot of people think they can see less or more of themselves as they move closer or farther away from a mirror.

When you look through a real window, the amount of scenery you can see on the other side does appear to increase or decrease as you vary your distance from the window. You can try this for yourself: Make a frame with your fingers and peer through it at arms length. Now bring the frame right up to one of your eyes, and watch as your field of view through your fingers expands dramatically. [Image via Shutterstock]

Why doesn't this happen as you approach a mirror? Well, it actually does. As you get close to a mirror, the amount of the virtual world you see reflected back at you will appear to increase. Similarly, as you back away, your window on the virtual world will appear to shrink. So why doesn't your view of yourself change? Because, unlike the virtual world, which remains fixed in place as you vary your distance from the mirror, your virtual self moves with you.

Let's go back to the window analogy. Imagine you're standing face-to-face in an open field with a friend the same height as you. Between you stands a brick wall, and in its center is a window with a vertical opening equal to half your height. You step back, away from the opening. If your friend does not move, the extent of her body that you can see will decrease as your window on the world beyond the wall appears to shrink. If, however, your friend steps back from her side of the wall at the same rate that you step back from yours, your bodies will remain fully visible to one another.

## Previous Weeks' Puzzles

### DISCUSSION

Catalyst123

First, I would like to point out that how you define a 'step' changes how many 'steps' there will be. I chose to define my steps as either filling or dumping the glasses, with pouring from one to the other being part of that step.

The solution for all three starts the same: Use the smaller glass to fill up the larger one, and use the remainder to get to the solution.

Ia) Use the 4oz glass to fill up the 9oz glass. This will leave 3oz in the 4oz glass.

Ib) Dump the 9oz glass and pour the 3oz into it.

Ic) Use the 4oz glass to again fill the 9oz glass. This should leave you with 2oz in the 4oz glass.

Id) Dump the 9oz glass and pour the 2oz into it.

Ie) Fill up the 4oz glass and pour it into the 9oz glass, giving you 6oz.

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IIa) Use the 7oz glass to fill the 11oz glass. This will leave 3oz in the 7oz glass.

IIb) Dump the 11oz glass and pour the 3oz into it.

IIc) Use the 7oz glass to again fill the 11oz glass. This should leave you with 6oz in the 7oz glass.

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IIIa) Use the 3oz glass to fill the 5oz glass. This will leave you with 1oz in the 3oz glass.

IIIb) Dump the 5oz glass and pour the 1oz into it.

IIIc) Fill the 3oz glass and pour it into the 5oz glass. This will leave you with 4oz in the 5oz glass.

I think I can do the first puzzle in less steps, but I haven't figured it out yet. Though I do have a solution where I fill the 9oz glass first, but it's just as many steps or more as the posted solution, depending on what I choose to call a 'step'.