stigant
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I don't know how many people like to do sudoku variants, but considering that several people like Ken-Ken puzzles, I thought I'd post some other sudoku type puzzles here. I've written a few puzzles in the Hidden Skyscraper Sudoku variant.

Rules:
Standard Sudoku Constraint - Fill the squares with the digits 1-9 so that each digit appears once in each row, column, and 3x3 region.
Hidden Skyscraper Constraint - A grid square contains an arrow(s) if and only if the number in that square correctly indicates the number of skyscrapers visible in that direction. (This is the modification from the mock test: if there is no arrow in a particular direction in a particular square, then the number of skyscrapers visible in that direction from that square is NOT the number on the square.)









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[Last edited by stigant at 04-26-2009 04:33 AM]

zex20913
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Does the number visible include potential skyscrapers in the selected square?

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stigant
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I'm not quite sure what you mean, but here's an example row:

728916534

Standing on the 2 square, looking right, you can see the 8 and 9 skyscrapers, so there would be a right arrow on that square, but not a left arrow as you can only see the 7 looking left. The 1 square would have left and right arrows, the 3 and 4 would have left arrows. No other squares on that row would have arrows pointing left or right (but could, of course have arrows pointing up or down, but we don't have the other rows to check this out).

Does that answer your question?

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RoboBob3000
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I think I'm unclear on the definition of "skyscraper".

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zex20913
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Is it that an arrow is pointing in the direction of the largest sequence of increasing numbers (relative to the current number) in that direction, unless there's a tie, in which case there's an arrow in both directions?

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stigant
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Robobob:
Think of the numbers that (will eventually be) are in each square as indicating the heights of buildings (skyscrapers). If you stand on any of the squares and look in one of the cardinal (NSEW) directions, you will see some of the buildings in that direction, but others will be obscurred by taller buildings between you and they. For example, if the buildings on a row have the following heights:

728916534

Then if you stand on the square with the 7-tall building and look East, you will see 3 buildings: the 2, 8, and 9 buildings, but the 9 building will obscure the rest of the buildings in that direction. If you stand on the square with the 3-tall building and look West, you will see 3 buildings (5, 6 and 9) but won't be able to see the 1 building (which is obscured by the 5 and 6 building) or the 7, 2, and 8 buildings (which are obscured by the 9). Since the number of buildings visible from the 7, looking right is NOT 7, there wouldn't be a right-arrow on that square. Since the number of buildings visible from the 3 square, looking left is exactly 3, the 3 square would have an arrow pointing to the left.

Zex:
Arrows are on all squares in which the digit in that square correctly indicates the number of visible buildings in that direction. If you can see the correct number of buildings in two or more directions then there will be two or more arrows on that square. Perhaps another example:

543219876
...>>....

There would be right arrows on the 2 and 1 since they can see 2 and 1 buildings to the right respectively. No other arrows would be on any of the other squares since those squares do not correctly indicate the number of buildings visible from themselves. (The 5 can see 2 looking right, 0 looking left; the 4 can see 2 looking right, 1 looking left; the 3 can see 2 looking right, 2 looking left; the 2 can see 3 looking left; the 1 can see 4 looking left; the 9 can see 5 looking left and 1 looking right; the 8 can see 1 to either direction, the 7 can see 2 left and 1 right; and the 6 can see 3 left and 0 right)

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[Last edited by stigant at 04-25-2009 12:45 AM]

agaricus5
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stigant wrote:
For example, if the buildings on a row have the following heights:

728916534

So what happens if you look east while standing on the 2-tall building? Can you see skyscraper 9?

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TripleM
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stigant wrote:
I'm not quite sure what you mean, but here's an example row:

728916534

Standing on the 2 square, looking right, you can see the 8 and 9 skyscrapers, so there would be a right arrow on that square

stigant answered that above.

One thing I'm still not entirely sure about though; if we had:

..853.. 

the 8 can see the 3, but

..852..

the 8 cannot see the 2.
Is that correct?

No wait, I'm still confused. In 289 you said the 2 can see the 9, but with 987 the 9 cannot see the 7. If we're using direct line of sight to make the second one work wouldn't the 2 not be able to see the 9?

  .
 ..
 ..
 ..
 ..
 ..
o..
...
...

(if I'm standing at the 'o' I can't see the building 9 tall.)

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[Last edited by TripleM at 04-25-2009 02:15 AM]

agaricus5
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TripleM wrote:

stigant wrote:
I'm not quite sure what you mean, but here's an example row:

728916534

Standing on the 2 square, looking right, you can see the 8 and 9 skyscrapers, so there would be a right arrow on that square

stigant answered that above.

Urgh. I don't know what I was reading that as. Clearly my brain is a bit sCraMBleD right now.

Oops.

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stigant
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Ok, I apologize: skyscraper puzzles are fairly well known among logic puzzlers, so examples are usually enough to illustrate/jog people's memories. Let me see if I can create a slightly more mathematical definition of what "see" means in this puzzle:

Let s(1), s(2), s(3), ... s(n) be digits in a row/column.
Let i < j. Looking right from s(i), you can "see" s(j) if for all k such that i < k < j, s(k) < s(j).

In other words, you can "see" a building if there are no intervening buildings which are taller than that building. Notice that the height of s(i) doesn't affect how many skyscrapers you can see from s(i).

I know that doesn't exactly jive with physical reality (which would require you to reason about slopes of buildings and your actual elevation etc) but it makes for a nice logic puzzle.

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stigant
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There were two errors in the first puzzle. I've fixed and re-posted the image. For reference (if you've already printed out the boards), the changes were:

R2C5 - removed the right-facing arrow
R6C1 - added a right-facing arrow.

I'm sorry for the inconvenience.

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