Author Topic: Knight Challenge -- try to move to each of the 63 remaining sqares... (Read 2854 times)

Neil Obstat · « **on:** July 09, 2016, 08:59:53 PM »

...without moving to the same square twice.

https://www.mathsisfun.com/games/knights-move-2.html

chrstnoel1 · « **Reply #1 on:** July 10, 2016, 04:06:25 AM »

Quote from: Neil Obstat

...without moving to the same square twice.

https://www.mathsisfun.com/games/knights-move-2.html

Thank you Neil. Interesting site.

Neil Obstat · « **Reply #2 on:** July 11, 2016, 09:57:52 PM »

Maths is fun is a great site for learning math, too.

They have a section on statistics that's very helpful for anyone having trouble understanding standard deviation, variance or deviation from the mean.

Or their treatment of algebra puts a new light on the topic that helps to understand it better.

If anyone wants to know how to do the knights challenge, I have some interesting pointers you can try out. I'm getting all the squares about half the time using certain patterns.

Neil Obstat · « **Reply #3 on:** July 13, 2016, 10:33:30 PM »

Someone calculated all the possible solutions for this (mark all the 64 squares with only one move to each square) and it was a pretty big number. On the other hand, the number of ways you can fail to keep the knight free, that is, by running out of places to move before hitting all 63 squares ahead of you, is astronomical.

In case I forget about this thread and someone reads it later, I'll just put down here a few helping hints to make success easier to achieve.

To get familiar with the general pattern that knight moves make on a chessboard, try moving close to the edge of the board all around, about 3 times, always moving say clockwise (you can go counterclockwise if it makes you more comfortable).

When you get to a corner square, you will quickly discover that if there is not still one of the two access squares left for your escape, you'll be stuck. Therefore, whenever you land on one of those two access squares you must move to the corner next and then leave by the second access square lest you be stuck in the corner.

When beginning with the knight in the a8 position, as this utility does, the squares c3 and f6 are likely to become a problem later on, so it's very helpful to move to those along the way during these first 3 laps around the board.

At some point near the end of 3 cycles you can begin to venture into the center area of the board, but it is very helpful to make circular loops, first clockwise, then counterclockwise, and clockwise again, always keeping in mind that you want to leave yourself with a pattern of unused squares that will be accessible by the knight as you start to run out of places to move.

The ending of your tour does not have to be a pretty form or arrangement, but if you can manage to leave a memorable pattern then it might be easier to remember how you did it so you can repeat it in the future.

The knight move is really strange for most players first starting to learn chess, because it's not easy to anticipate where the knight will be able to move after 4, 5 or 6 moves, unlike a bishop or a rook, which have straight line powers. This puzzle is an excellent way of becoming more instinctively familiar with the powers of the knight, and becoming skilled at solving this simple puzzle cannot but increase your ability to move your knight well in a game of chess.

Neil Obstat · « **Reply #4 on:** July 14, 2016, 05:30:32 PM »

Okay, this is it.

I'm thinking if I don't post this solution, I'll eventually forget how it goes, so if I post it here then I can always come back here and find it, providing I have Internet access that is.

So, this is following the guidelines I gave above, plus a few nuances I didn't manage to mention before; namely these:

1) The knight can move in a 3x3 square donut pattern, landing on all 8 contiguous squares in succession. But the center square is not accessible with this 3x3 figure, unless the knight moves outside the square, and it has to move to a very select few of the 16 squares that surround the 3x3 figure. Therefore, when choosing where to move, and one of these 3x3 figures shows up, it is advantageous to keep in mind how you are going to hit that center square, lest you end up stuck there or left unable to reach it.

2) In the following scheme, the squares c3 and f6 are quite often left over with no way of reaching them. Therefore, it is advantageous to take the opportunity to hit them early in the sequence, even if it causes a slight disruption to your scheme. (Actually, I did mention this nuance above.)

3) There are key moves where you have options to move differently if you so desire, but if you deviate in a way that does not include sufficient foresight it could very likely be the cause of finishing incompletely. To make this scheme, below, more easily understood, I have put in BOLD those squares to which this principle applies. That is to say, the bold squares are where the pattern is disrupted.

a8 - e3 in 63 moves:

a8 c7 e8 g7 h5 g3 h1 f2
d1 b2 a4 b6 c8 e7 g8 f6
h7 g5 h3 g1 e2 c1 a2 b4
a6 b8 d7 f8 g6 h8 f7 h6
g4 h2 f1 d2 b1 a3 b5 a7
c6 a5 c4 d6 e4 c3 d5 f4
d3 e5 f3 h4 g2 e1 c2 a1
b3 c5 b7 d8 e6 d4 f5 e3

In case it isn't obvious, these are algebraic notation terms for an 8x8 chessboard indexed as follows:

8 | | | | | | | | |
7 | | | | | | | | |
6 | | | | | | | | |
5 | | | | | | | | |
4 | | | | | | | | |
3 | | | | | | | | |
2 | | | | | | | | |
1 | | | | | | | | |
a b c d e f g h

Bonum ad omnes · « **Reply #5 on:** July 14, 2016, 05:44:16 PM »

Quote from: Neil Obstat

...without moving to the same square twice.

https://www.mathsisfun.com/games/knights-move-2.html

When you found this did you try it? and if so, what was your score/results? Or did you work out the equation first?

Just curious. :D

Neil Obstat · « **Reply #6 on:** July 14, 2016, 07:20:08 PM »

Quote from: Bonum ad omnes

Quote from: Neil Obstat
...without moving to the same square twice.

https://www.mathsisfun.com/games/knights-move-2.html

When you found this did you try it? and if so, what was your score/results? Or did you work out the equation first?

Just curious. :D

I first attempted this challenge about 40 years ago, using a chessboard and pistachio shell halves to mark the squares I had moved to. I guess pawns would have been easier to use but I didn't have 63 pawns handy at the time.

I gave it a shot from time to time over a year or two and never managed to get to the end without having to double dip on at least one square. I recall having read in books that it is possible, though, so I never forgot about it.

A few months ago, I noticed this feature on Maths Is Fun and didn't try it because I had been frustrated with it long ago (like in 40 years ago). But just the other day, I had a little time and gave it a shot. The computer option is far superior to keeping track using a physical board because the latter practically necessitates using two hands, one to move the knight and the other to place a marker on the square evacuated. It's really distracting to do that. But with a computer program you get a lot of continuity, and the memory of what pattern you tried last time and the time before and the time before, etc., up to twenty or thirty attempts, is far more easily retained than when you spend 5 or more minutes doing it manually. With the computer, 30 seconds is often enough, and if you find yourself stuck halfway, 15 seconds may suffice. So you get a lot of practice in a short time, and you learn faster that way.

In answer to your question, when I first tried it a couple weeks ago, I got very close to success right away, but it did take about a day or two (2 hrs per day) before I finally succeeded, and it was a great feeling of victory when I did. I was so thrilled I had to share it somewhere, so CathInfo was my victim!

I was getting scores of 65 to 70 quite often, but a few 64. Finally I got 63, and then using the principles I outlined above, I was able to get 63 again and again, about half the time, along with 64, 65, 66 and 67.

As for "equation," I never really discovered what the mathematical formula is for the solution, although I expect there must be one, somehow. It's a very logical and structured sequence so I'm sure it's possible to set up an equation of some sort. But I have no clue whatsoever how to do that.

BTW -- the color of the squares might make a difference for some players; so that you know, the accepted norm for chess boards is: square a1 is black and square h1 is white.