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# The ten-thousand second Pyramid

OK, gang. If you've been vaguely interested in croco-puzzle.de based on my recent description and my recent walkthrough, but a sense of "but these puzzles are going to be so hard, I'm not going to be able to do them" is putting you off, today's daily puzzle may be a good starting-point. It's only going to be available (at least, as a puzzle that counts towards the overall ratings) until 23:59:59 German time tonight, or 22:59:59 British time, 17:59:59 Eastern time or 14:59:59 Pacific time, now the US clocks have gone back as well as the US ones, so about five and a half hours' time. Today's rated puzzle is a Pyramid, which is a type of puzzle that (with only one small exception) I've only ever seen at croco-puzzle, so here's an introduction to this type of puzzle, starting with the very basics.

Pyramid puzzles are based around a triangular array of squares, some of which will be filled in already.

1) Fill in a digit from 1 to 9 in each of the blank squares. No zeroes, no hexadecimal.
2) The digit in each square has to be either the sum or the difference of the two digits in the squares below it.
3) Some rows are grey, and all the numbers in the squares in a grey row are different to each other.
4) Some rows are white, and at least two numbers in the squares in a white row match each other.

So if you have boxes like this: ...then you can fill the top box in as there is only one possibility; the sum of 4 and 9 is not a single digit, so the top box must be the difference of 4 and 9, so 5.

However, if the boxes look like this: ...then you can't fill in the top box yet, as it could either be a 9 (the sum of the two digits beneath it) or a 1 (the difference of the two digits beneath it).

Applying a similar principle to a very slightly more complicated example: The colour of the row actually makes a difference at this point. We can fill in the left-hand blank box uniquely at this point; as the sum of 8 and 2 is not a single digit, the box above the 8 and the 2 must be the difference, 6. The number above the 2 and the 4 could either be the sum, 6, or the difference, 2, but as the row is grey, the digits in the row must be different. Accordingly, the second digit cannot be a 6 and so must be a 2.

I will leave the rest of these to you. They start barely above trivial and get slightly more difficult. Select the following text for hints: use the same logic as for the previous puzzle. We can solve the right-hand digit uniquely, then the fact that the row is grey forces the left-hand digit. Select the following text for hints: use the same logic as for the previous puzzle, but use the fact that the row is white, and so must contain at least two digits the same, rather than grey. Select the following text for hints: there are two different possibilities for each digit, but only one combination of the two possibilities will permit the top row to be correct. Select the following text for hints: this one is a little harder. One digit on the second row can be determined uniquely first, then you can identify two different possibilities for the other two digits on that row. However, only one combination will permit the top row to be filled legally because it's a white row. Select the following text for hints: this one is a little harder still. Two digits in the third row can be determined uniquely on their own, then work through the possibilities for the third digit in the third row. There may appear to be some different possibilities for the second row, but remember the meaning of its colour and what the top row means. Select the following text for hints: this last one is a little harder still. Remember the meanings of the grey rows and the conclusions that this forces.

You'll probably pick up some tips for yourself. For instance, 1s must be the difference between the digits beneath, 9s must be the sum of the digits beneath, a 2 can only be the sum of two 1s if the row below is white (otherwise it must be a difference), a 8 can only be a difference of a 9 and a 1, that sort of thing.

Today's puzzle is more like the last one than any of the others, but on a slightly larger scale. You can start by making some logical deductions, but there may come a point where you just have to try some possibilities, follow the logical conclusions of your trial, then if your trial forces two identical digits on a grey row, all-different digits on a white row or a set of digits that cannot work with those above them, then you will have to take steps backwards. Use the different colours for trials and you can use the Ruckgangig facility to remove all the boxes filled in with one particular colour. This is a fairly high-variance sort of puzzle; if your first trial turns out to be the correct one, then you may work through very quickly, if not then you will have to identify the backsteps and retry things as efficiently as possible.

I was fortunate enough to get lucky and wound up with a very good time, proving a faster solver than, well, some people who are considerably better puzzle-solvers than me. Today's fastest solver - to date! - has been a remarkably low-ranked solver, compared to most "fastest solvers of the day". I really enjoyed today's puzzle and think it's a very good place to start.

So if this has tickled your fancy, follow my walkthrough if you haven't done so already and consider giving today's puzzle a try! Beware, you may grow to enjoy having the crocodile bite its teeth into you...

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