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Hi Vietnamese, and welcome to the forum!

I found a trig solution...see image below, which hopefully is clear enough. If not, just let me know and I'll try to explain.

I've shown a 3.2-wide tape starting at bottom left, moving upwards to the right (as indicated by the arrows) and wrapping around the vertical pipe 1.5 times. The broken lines show the location of the tape behind the visible pipe face.

The figures I've used are just examples, so enter your own into the formula to suit.

Formula to find angle GAH:

This next formula is shorter and will give exactly the same answer, but I think the proof might be a bit longer:

To ensure that the tape will cover the entire vertical pipe face, the top left corner of the tape must start at or below the bottom of the pipe.

Roman numerals (C/C/I):

ONE:

400:

169,196,961:

9+0:

9-8:

10-9:

81-0:

6^0, 8^0, 9^0, 10^0:

6/6:

8/8:

9/9:

Hi;

Three green matches are relocated to form a '4' in front of the red '0', and one green match is removed.

This forms the number 400, which is a square (20 x 20).

Inspired by Bob's idea of nine being a square number.

Hi Bobby;

Everybody has a twin somewhere in the world

Very limiting, that is.

Marsling and earthling twins, first revealed and announced here on MIF!

I've never used a telescope.

I once had a Canon TTL camera with a 75-210 zoom and a 3x televerter that gave me 630x, and I set it up on a tripod to view a comet's tail. However, it was too cloudy and I saw nothing...and I froze.

I should have tried again, but haven't really had the interest and now I don't have that camera anymore.

Off to bed now...catch you later.

I have a Sony camera with a little telescopic lens, and an imagination that fills in any missing information that my camera may not have picked up for one reason or another.

I don't know how good the viewing is from here. All I know is that viewing the heavens is best done at night, but that's when I'm either asleep in bed or asleep at my computer, so I don't get to see much then.

No. I didn't see any canals through my telescope, either, just like Mariner 4 didn't.

I was just reading up on him...

Later, with notable thanks to the observations of the Italian astronomer Vincenzo Cerulli, scientists came to the conclusion that the famous channels were actually mere optical illusions. The last popular speculations about canals were finally put to rest during the spaceflight era beginning in the 1960s, when visiting spacecraft such as Mariner 4 photographed the surface with much higher resolution than Earth-based telescopes, confirming that there are no structures resembling "canals".

Or had all the water run out of the canals and dropped off the planet, and all evidence of their existence eroded away, by the time Mariner 4 came along...?

Hmmm...interesting.

No, I hadn't heard of GS or JS, only Uncle Martin, but he doesn't look at all like our two guys. Maybe he's still evolving...or is it our two who are evolving?

bobbym wrote:

Here is my best, I call it, The Man from Mars.

This is a relative of his, I think:

Hi Leren;

Here are two options that, by making one change to the puzzle in post #6, converts it from unsolvable to one that has a unique solution:

```
Option 1: C1 empty Option 2: G1 empty
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
A | _ 2 _ | 1 5 _ | _ _ _ | A | _ 2 _ | 1 5 _ | _ _ _ |
B | _ _ 5 | 2 _ _ | _ _ 9 | B | _ _ 5 | 2 _ _ | _ _ 9 |
C | _ _ _ | _ _ 4 | _ 2 5 | C | 3 _ _ | _ _ 4 | _ 2 5 |
|-------+-------+-------| |-------+-------+-------|
D | _ 8 7 | _ _ 2 | 5 _ _ | D | _ 8 7 | _ _ 2 | 5 _ _ |
E | _ 4 _ | _ 1 7 | _ _ _ | E | _ 4 _ | _ 1 7 | _ _ _ |
F | _ _ _ | _ _ 3 | 1 _ 7 | F | _ _ _ | _ _ 3 | 1 _ 7 |
|-------+-------+-------| |-------+-------+-------|
G | 2 _ 4 | 6 _ _ | _ _ _ | G | _ _ 4 | 6 _ _ | _ _ _ |
H | 6 _ _ | _ 9 _ | _ 3 _ | H | 6 _ _ | _ 9 _ | _ 3 _ |
I | _ 5 _ | _ _ _ | 9 6 8 | I | _ 5 _ | _ _ _ | 9 6 8 |
```

Both solutions are easy to find, and are different from each other.

Hi, Leren (Dutch for 'learn'), and welcome to the forum!

Well, that's interesting information! I'd suspected that there would be many more solutions to the OP's puzzle (post #1) than the few that I'd found with the Excel Solver add-in I referred to in other posts, but not as many as that!!!

How did you arrive at that total, and what program did you use?

Hi;

The hour and minute hands are at equal distance from the 6 hour, what time will it be exactly?

There are 12 solutions where the two hands are on different sides of the clock face, and the distance to the 6-hour mark for one is measured in the reverse direction to that of the other.

There are 11 solutions where one hand is on top of the other and the direction to the 6-hour mark for both hands is the same.

EDIT: I've had another think about this, and I reckon that although the puzzle wording doesn't say so, we're probably just meant to give the exact time of the clock hands shown in the image on the puzzle page (and reproduced below)...in which case there's only one solution. It doesn't really make sense otherwise, there being so many solutions.

Hi, SynergySmith, and welcome to the forum!

I have an answer:

Sorry...I should go now. Big day coming up for me tomorrow.

Catch you later.

Witness the story of Andrew Wiles...

Wow! What persistence!! I just read the Wikipedia article.

My Excel T&E only took a few seconds longer than a few seconds, which I was quite pleased about.

Even at my age I don't have centuries to wait for your 100-page proof to appear - and besides, I have to go to bed soon - but I'll hang around here for a bit longer in case you can get your idea to work.

All the best with it!

I thought they were synonyms, but you've now thrown some doubt on that.

Do you have any idea about how to solve this mathematically?

bobbym wrote:

You could possible try to enter it at the wise and all powerful oracle at Wolfram Alpha. If you do, let me know what happens.

No solution, but scads of nutritional info!!

Hi Bobby;

That works perfectly...thanks! I've never used Reduce before...hope I can remember to use it if the opportunity arises again.

Hi chen.aavaz;

bobbym's code is Mathematica, which we both have. He knows it very well, and I know some.

chen.aavaz wrote:

I don't know though how to express all these in a mathematical way.

Same here. I started on this puzzle in much the same way as what you've shown, but couldn't progress from there to a mathematical solution.

That's when I turned to T&E in Excel for the solution, and for confirmation that it was unique.