Without magnets and glue, how can they be combined?
UINVEA
Silicon Forest
If the type is too small, Ctrl+ is your friend
I play Wordle most every day, it's just my speed. Some days the answer appears as if by magic, other days I am wrought with anguish. Today's puzzle was super-annoying. On my second guess I got three of the letters in the right place. Do you know how many words there are with the letters placed like this:
_ O _ E R (highlight to read)
I thought of nine right off the bat, and I've only got four places left to use. Well, what I need is a word made entirely of letters that haven't been used yet, but that doesn't help much. I've still got 16 letters to choose from, I may as well be starting blind. If I am lucky, sticking in some random word would help, but would it be enough help to be worth one of the four places I have left? I decided it wasn't and instead tried to eliminate as many words as I could from my list. Two words had a B and two words had an N, so with two guesses I was able to eliminate four words from my list. It wasn't enough. The answer was the name of a fictional TV persona.
(H O M E R) (highlight to read)
It was on my list, but I had discounted just because of the name association. Too late I realized it also referred to a baseball play, and it's spring, and all right thinking people's thoughts turn to baseball. If you're not thinking about baseball, that makes you some kind of commie. Wasn't that the standard test of suspicious persons during WW2 and the Cold War? Guess I flubbed that test.
Megaminx |
Got an email from GANCUBE promoting their line of Rubik's Cube puzzles. I thought they were a fad that faded away a while back, but evidently it's still going strong. These guys seem to have a near endless variety of these things. I've played with them a few times and I think I even solved one, once. I don't know why, but they never really grabbed my attention. They're cute and all, but what do you get when you solve it? The same thing you started with. And is the 'solution' any more valid than any other combination? I know most puzzles / games are the same way. Maybe they were too much work, maybe I'm not as smart as I think I am.
Sunday Jumble |
Vincent Vega (John Travolta) & Jules Winnfield (Samuel L. Jackson) |
I started watching this video the yesterday. I got as far as the square cross puzzle (timestamp 1:45 to 2:20), and that intrigued me. I stewed on it overnight and this morning I came up with this analysis.
Square Cross Problem
We start with a drawing of a cross. The cross is composed of five squares. The center of the cross is a square. Attached to each side of the central square is another square of the same size.
Our mission, should we choose to accept it, is to cut the cross into five pieces and then reassemble those pieces into two crosses with the same shape. These two new crosses are the same size as each other.
Since we are making two crosses from the original larger cross, each of the smaller crosses must be one half of the area of the original larger cross.
If the length of the side of one of the component squares of the larger cross is one, a little algebra will show us the the length of the side of one of the component squares of the smaller crosses must by the (2^0.5)/2 (the square root of two divided by two).
Cutting each of the component squares of the original square on both diagonals will give us 20 right isosceles triangles. Two of those triangles joined together along their hypotenuse will give us a square the same size as one of the component squares of the smaller crosses. Two triangles for each of five squares requires ten triangles to make one smaller cross. Two times ten is twenty, the number of triangles we got by cutting up the larger cross.
This is the brute force method. It shows we can cut up the larger cross into smaller pieces and reassemble them into two, smaller, crosses. However, we have twenty pieces, not five, so we do not have a solution to the stated problem.
The trick is to find cuts that do not need to be made so that when we cut up the larger square we only have five pieces. Now I'm thinking a computer program could make short work of this if I can just figure out how to encode it.
Completed Puzzle |
Dennis gave me this puzzle for my birthday. I spent a few minutes trying to solve it but then realized I could write a computer program to solve it, much like I did for another puzzle he gave me.
It didn't take long to write it, but it took a while to track down the bugs and to polish it up a bit.
Puzzle Pieces |
The first problem was how to represent the puzzle pieces in the program. With the cube puzzle, I was able to use a 64-bit integer to represent the 4 x 4 x 4 cube. This puzzle was a little trickier.
Seven Segment Display |
The digits here are modeled on the seven segment displays that are common in simple electronic devices. Note that each segment is shaped like a rectangle with a triangular point on each end. If the puzzle pieces closely followed the electronic pattern it would have been simple to encode them as data. However, some of the pieces, like the zero, one and seven, included a bit that isn't part of the original pattern, and some (the zero) were missing a bit.
Zero |
I decided to 'draw' the digits using ASCII characters. Each true segment was represented by three characters. I couldn't come up with a good visual method of representing multiple triangles in one space, so I gave each segment three full spaces and left a space between segments.
Playing Field |
Today's Jumble |
1943 Soong May-ling in White House Oval Office gave a press conference. |
Number of times search was blocked by level. |
Levels 0 thru 4 don't even register above the X axis at this scale. Including them made the key confusing, there wasn't enough differentiation in the colors, so I didn't include them. Level 5 shows a flat line even though it has counts consistently above 10 billion (10^10). I included it just to give you an idea of the scale we are working with.
The program printed this data to stderr, which by default goes to the terminal screen. Copy and paste that into the text editor and then replace all instances of multiple spaces with a single tab. (Start with the longest sequence of spaces you can find and then work backwards.) The formatting is lost, but now you can copy and paste that into a spreadsheet and our formatting is restored.
I did some calculations in the spreadsheet to see how many comparison tests we avoided and came up with about 4.5 quadrillion, which is a very large number, but still nowhere near the value of
12 * (3*96)^4 * (4*96)^5 = 6.8929848e+23
which is what I expected. So there is something wrong with my calculations somewhere.
P.S. Something weird happened when I pasted the text into the spreadsheet. I got a bunch of lines consisting of the single word #ERROR! I looked in my source code, but couldn't find it, and I just looked in the text file and it's not there either, so I think Google Sheets must have found something it didn't like (like a line of 300 periods) and complained. No problem, simple sort the file and all the ugly stuff you aren't interested in gets lumped together and can easily to erased.
P.P.S. One trick I use when dealing with large quantities of data is to insert a column along the left hand edge, a new column A if you will, and then number all of the lines with sequential numbers. You can easily do this by typing 1 in box A1, and then =A1+1 in box A2. Copy box A2, select all the rest of column A from A3 to the end paste. Now all the lines are numbered sequentially. However, you are not quite done. If you sort the sheet now, all the values in column A will be recomputed and your original order will be lost. So what you do is copy column A and the Paste Special -> Values Only back over it. Now you have an indelible original order. Sort however you like, but you can restore the original order by sorting on Column A.
Recursion |
I modified the program so it now finds all 335 solutions. It runs quite a bit longer now, almost 15 hours. I also uploaded a text file that illustrates all how all the transformations are done to place a piece in the cube.
The program made 41 billion comparisons, which is a bunch, but a far cry from the umpteen zillion I expected. Since the 10th root of 41 billion is only about 11, I'm thinking most searches must have reached the end of their list early on.
The program starts with the first piece placed in one of it's twelve possible positions. The first piece has only 12 possible positions as all of the other positions (276) are simply rotations of one kind or another.
We are going to test each of these against all 288 possible positions for the 2nd piece. Some of these tests will fail, indicating the pieces would collide, but most will pass. Of those that pass, we will test all 288 positions for the 3rd piece. More tests will fail at this level and fewer will pass. There will be 335 searches that will go through all of possible positions at all levels, but that only amounts to
(12 + (3*96)*4 + (4*96)*5) = 3084
I suppose I could track how deep each search goes and perhaps a graph of the averages might give some insight, but I expect it's some kind of exponential curve. Besides, I got my solution, put the puzzle together and shipped it off to my niece. We shall see what she does with it.
Dennis made this puzzle. It got taken apart a while back and I never mustered the drive to put it back together, but now I want to send it to my niece, so I really should put it together so she knows it can be solved. I don't know how long solving it by hand would take, but it should be a simple matter to write a computer program to solve it, and it was. Took me a few hours over the last couple of days.
In a single plane:
A piece can be in any of the 4 planes.
The cube can be sliced into four planes and this can be done 3 ways:
Multiply those together (4 x 2 x 4 x 3) and we have 96 possibilities.
In a single plane, using a single orientation, just sliding the piece around in the 4 x 4 space:
In sum:
Combining all these we have (3*96)^5 x (4*96)^5 or 1.6543163e+25.
I think. I may have it all backwards.
If that is the correct value, it is doubtful whether a brute force program will be able to solve the puzzle. I'm guesstimating it will take a billion years. On the other hand, many possibilities will fail almost immediately. At this point I was wondering whether it is worth going ahead with writing the program.
The lure of solving an unsolvable problem was too great and I went ahead with the program. It only took a few hours over a couple of days to write and after 7 or 8 minutes of execution time, it came up with a solution that actually worked.
Pipe Organ Desk |
2 + 3 = 10
7 + 2 = 63
6 + 5 = 66
8 + 4 = 96
Then:
9 + 7 = ?