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LAP »  Forums » Reviews

Subject: A neat puzzle-solving game that only needs graph paper and pencils. rss

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Russ Williams
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No forum posts so far about this game? It deserves some attention!

LAP is a neat little paper and pencil puzzle game by Lech Pijanowski , a Polish game enthusiast who wrote a large book of various board games, which was influential in its day and introduce many games to a wider audience in Poland. He also designed some games, including LAP: the title comes from his initials. The game achieved a wider audience when Sid Sackson described it in A Gamut of Games .

LAP is sort of cross between Battleship & Mastermind & Polyominous/Fillomino puzzles :

Like in Battleship, both players try to deduce a 2-dimensional arrangement created by the opponent. Thus it’s not really an interactive game but more of a simultaneous solitaire activity, like Battleship at a high level.

But like Mastermind, LAP involves interesting nontrivial logical deduction at a level higher and more challenging than that of Battleship.

Like Polyominous puzzles, it includes reasoning about contiguous sections of a given size.

RULES:

Concretely, both players secretly mark 4 regions of 16 squares each on their hidden 8×8 grid. Then they take turns asking about a given 2×2 section of the opponent’s grid. E.g. in the sample arrangement shown here, asking about “AB12” (the upper left corner) would receive an answer that “3 of the squares are in section I and 1 of the squares in in section II” (or “1 1 1 2” for short). The answer does NOT indicate which square is which, only the number of squares which come from each of the 4 sections. (Thus I prefer to say the 4 squares in sorted order from lowest to highest.)

Eventually one of the players figures out the opponent’s diagram and declares a win. Of course if it turns out the player is wrong, then the player loses!

Both players get an equal number of turns, so that they could theoretically both declare on the same turn, resulting in a tie if they are both right or both wrong. I’m guessing that’s not common in practice.

We also played it 3-player once, with each player guessing the diagram of the neighbor on their left. That worked fine.

GAMEPLAY:

Depending on the difficulty of the diagram, the skill of the guesser, and a bit of luck on the guessed quadrants, one gradually becomes certain about more and more of the secret diagram and (in my limited experience so far) can usually determine it after 20 or so questions.

There is no prescribed manner for noting your gathered info. It seems a good practice for both players to keep a running log of the questions and answers, to keep track of whose turn it is (occasionally there are thinky pauses) and to double-check past answers and be able to ascertain if incorrect info was accidentally given (this happened once to us, alas!). Beyond that, different players prefer different methods of keeping track of what they know.

STRATEGY:

Of course you want to create a diagram that’s hard to solve. So something like

1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4

or

1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2
3 3 3 3 4 4 4 4
3 3 3 3 4 4 4 4
3 3 3 3 4 4 4 4
3 3 3 3 4 4 4 4

will be too easily guessed. It seems good to reduce the number of 2×2 regions that are all the same section. Without too much trouble, you can make an irregular diagram that has only 5 or 6 monosection 2×2 regions. On the other hand, making an easier diagram could be a useful handicap technique when playing with a newbie or child.

When solving, I like to get information about the 4 corners soon. But I’m still learning about how to solve well and efficiently. At first we tended to start with AB12, AB34, CD12, etc, i.e. “tiling” the 8×8 diagram with 2×2 regions, but soon we began experimenting with overlapping rather than nonoverlapping 2×2 regions as well.

The fact that each of the 4 regions has exactly 16 contiguous squares is of course crucial and helps you deduce things with a variety of types of reasoning as you get more information.

For example, suppose you discover that the upper left corner contains 1 1 1 2 in some arrangement. Then you know the upper left corner must be 1, because if it were 2:

2 1 .
1 1 .
. . .

then that 2 in the upper left corner is isolated, obviously cut off from being part of a contiguous group of size 16.

If you enjoy solving “this kind of puzzle”, I think you’ll enjoy working out an opponent’s diagram.

RECOMMENDATION:

The rules are quite simple, but depending on your experience with solving “this kind of puzzle”, you might find the gameplay to be annoyingly difficult.

If you simply dislike “simultaneous solitaire” games of any sort on principle, you will be annoyed that the gameplay is, well, not “gamey” or interactive at all.

Personally I like this kind of puzzle. And my wife and I have been known to solve the same sudoku puzzle simultaneously & independently in competition. Whether that’s technically a “game” or not, I don’t know, but we enjoy it. LAP gives us a way to do that sort of thing, but with the added fun of easily making our own puzzles that are interesting for the other to solve.

So if you think the type of puzzle in LAP sounds interesting, or if you’re already a fan of Mastermind or Battleship or Polyominous/Fillomino puzzles, then I highly recommend trying LAP. All you need is some graph paper and pencils!

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  • Last edited Tue Oct 18, 2011 1:03 pm (Total Number of Edits: 2)
  • Posted Tue Oct 18, 2011 12:12 pm
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Larry Levy
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What a fantastic game this is! I probably played this at least 100 times with my mom when I was a teenager and we both loved the mental workout it gave us. It was definitely one of my inspirations when I created Deduce or Die ; I was trying to replicate a similar kind of reasoning with a game that was closer to Clue rather than Battleship. Fans of logic puzzles should drop what they’re doing, grab some graph paper, and start a game right now. You won’t be disappointed!
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  • Posted Tue Oct 18, 2011 5:15 pm
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Thumbs up not only for teaching me about a game I hadn’t heard of before, but for including a link to my blog.
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  • Posted Sun Dec 4, 2011 1:38 am
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Grant Fikes
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Question!

russ wrote:
Eventually one of the players figures out the opponent’s diagram and declares a win. Of course if it turns out the player is wrong, then the player loses!

What if I use this grid?

11111111
12121212
12121212
22222222
33333333
34343434
34343434
44444444

As I understand the rules, no amount of information can narrow the grid down beyond four possibilities. This means that my opponent has only a 1 in 4 chance of winning if he tries to guess my board.

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  • Last edited Sun Dec 4, 2011 5:42 pm (Total Number of Edits: 1)
  • Posted Sun Dec 4, 2011 5:42 pm
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mathgrant wrote:
As I understand the rules, no amount of information can narrow the grid down beyond four possibilities. This means that my opponent has only a 1 in 4 chance of winning if he tries to guess my board.

A very interesting question indeed! I’d not even thought about such pathologically ambiguous arrangements. Hmm. As far as I can tell, it looks like you are right, that this example has 4 possible solutions for the information the opponent could receive. Assuming we are not both brain-farting, I wonder if the author knew the game had this “feature”.

Perhaps I will try that arrangement the next time I play… whistle

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  • Posted Mon Dec 5, 2011 1:32 pm
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Bart Wright
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mathgrant wrote:
Question!

russ wrote:
Eventually one of the players figures out the opponent’s diagram and declares a win. Of course if it turns out the player is wrong, then the player loses!

What if I use this grid?

11111111
12121212
12121212
22222222
33333333
34343434
34343434
44444444

As I understand the rules, no amount of information can narrow the grid down beyond four possibilities. This means that my opponent has only a 1 in 4 chance of winning if he tries to guess my board.

I noticed this possible class of patterns 35-odd years ago and have wondered if anyone else had noticed it too, but until today I’d never found a forum where the game was discussed (not that I looked all that hard). I’ve been playing Lap for years, ever since I got the book, though for me it’s hard to get opponents. You can imagine expanding the rules to prohibit ambiguous patterns, or allow 4×4 blocks that go off the board (a 2×1 or 1×2 or 1×1 only at the edge). But I also found that humans can make patterns that are frustrating in any case with lots of little parapets of castles…

But I wrote a program back in the late 1980s so I can play it solitaire (I guess its pattern; it doesn’t guess mine), and since the program’s not trying to trick me, it’s actually more fun. It also never makes mistakes in answering my questions, which all too many humans do. I still have the .exe file, which runs in a DOS command-prompt window, which I’d be happy to share. I don’t know enough about this forum to know how or if you can share .exe files, which can of course be notorious carriers of viruses.

My preferred strategy is to start with a 9-square pattern of guesses covering the inside of the board completely excluding the outside edges, so bc23, de23 … fg67. Once you get the center, the edges are often much easier.

I also found a pattern for which, if you make a correct set of four guesses, you can uniquely determine the pattern.

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  • Posted Mon Dec 19, 2011 6:52 pm
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Bart119 wrote:
I also found a pattern for which, if you make a correct set of four guesses, you can uniquely determine the pattern.

!

Intriguing…! Now I must ponder…

 
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  • Posted Mon Dec 19, 2011 7:53 pm
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Grant Fikes
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I think I found the 4-clue solution!

Spoiler (click to reveal)
12345678
A22222444
B21332444
C21332244
D21333244
E21333244
F21333244
G11333114
H11111114

AB23: 1223
FG67: 1124
GH12: 1111
GH67: 1111

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  • Posted Wed Aug 8, 2012 5:00 am
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mathgrant wrote:
I think I found the 4-clue solution!

Spoiler (click to reveal)
12345678
A22222444
B21332444
C21332244
D21333244
E21333244
F21333244
G11333114
H11111114

AB23: 1223
FG67: 1124
GH12: 1111
GH67: 1111

Hmm… I tried solving it with your 4 clues and got something similar but not identical!

Spoiler (click to reveal)
12345678
A22224444
B21322244
C21333244
D21333244
E21333244
F21333244
G11333114
H11111114

It seems there are some “miai” near the top edge…

Am I brain-farting, or are there indeed 2 possible solutions consistent with the 4 clues?

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  • Posted Wed Aug 8, 2012 11:48 am
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D’oh. At least I found a five-clue solution, which is a vast improvement over my previous record of seven (where the regions are all 4×4 squares).

Edit: How about this?

Spoiler (click to reveal)
AB67: 1111
BC78: 1223
EF45: 1233
GH12: 1111
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  • Last edited Wed Aug 8, 2012 12:09 pm (Total Number of Edits: 1)
  • Posted Wed Aug 8, 2012 11:58 am
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mathgrant wrote:
D’oh. At least I found a five-clue solution, which is a vast improvement over my previous record of seven (where the regions are all 4×4 squares).

Edit: How about this?

Spoiler (click to reveal)
AB67: 1111
BC78: 1223
EF45: 1233
GH12: 1111

I found this solution:

Spoiler (click to reveal)
[c]
44444112
44444112
44413332
44413332
41113332
41223332
11233332
11222222[c]

EDITED: see below for correction of transcription error in this.

(But I wouldn’t bet my life that it’s unique! See if it’s the same as yours… it looks promising at least!)

 
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  • Last edited Wed Aug 8, 2012 1:07 pm (Total Number of Edits: 2)
  • Posted Wed Aug 8, 2012 12:18 pm
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russ wrote:
mathgrant wrote:
D’oh. At least I found a five-clue solution, which is a vast improvement over my previous record of seven (where the regions are all 4×4 squares).

Edit: How about this?

Spoiler (click to reveal)
AB67: 1111
BC78: 1223
EF45: 1233
GH12: 1111

I found this solution:

Spoiler (click to reveal)
[c]
44444112
44444112
44413332
44413332
41113332
41223332
11233332
11222222[c]

(But I wouldn’t bet my life that it’s unique! See if it’s the same as yours… it looks promising at least!)

Your 1’s aren’t connected, and you have too many 4’s, but the obvious fix matches my solution.

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  • Last edited Wed Aug 8, 2012 12:35 pm (Total Number of Edits: 1)
  • Posted Wed Aug 8, 2012 12:34 pm
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mathgrant wrote:
Your 1’s aren’t connected, and you have too many 4’s, but the obvious fix matches my solution.

Oops, yeah, I goofed when transcribing from my paper solution, erroneously “saving time” with some copy and paste…

Spoiler (click to reveal)

44444112
44411112
44413332
44413332
41113332
41223332
11233332
11222222
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  • Last edited Wed Aug 8, 2012 1:07 pm (Total Number of Edits: 1)
  • Posted Wed Aug 8, 2012 1:06 pm
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mathgrant wrote:
Question!

russ wrote:
Eventually one of the players figures out the opponent’s diagram and declares a win. Of course if it turns out the player is wrong, then the player loses!

What if I use this grid?

11111111
12121212
12121212
22222222
33333333
34343434
34343434
44444444

As I understand the rules, no amount of information can narrow the grid down beyond four possibilities. This means that my opponent has only a 1 in 4 chance of winning if he tries to guess my board.

Maybe a fix is to allow some number of “power guesses” that give you just a single square (say 2-4 per player). Could also be a means of handicapping the game.

 
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  • Posted Thu Nov 19, 2015 11:20 pm
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Thread necromancy? Oh, well.

Quote:
There is a pattern and a series of 4 guesses that allow determining the entire board, which I leave as a puzzle.

I guess I never gave my own answer, so I’ll give it here.

Spoiler (click to reveal)

11112222
13312442
13334442
13334442
13334442
13334442
13312442
11112222

The guesses are de23, de67, ef45, and ab12. The first 2 guesses with their “1234” require that regions 1 and 2 stretch around the edges to barely connect. The 3rd guess determines a clean boundary between the 2 central regions, after which you know the exact shape of the board — you just don’t know the names of 3 of the 4 regions! ab12 lets you determine that.

As I said before I wrote a program in the late 1980s to generate LAP boards. I have the .exe still but lost the source code along the way somewhere, but I can tell you the basic idea. I start with 3 random squares as the start of regions and add a single square to each in turn – and then check that all of the blank squares can still be connected to each other. At the end the 16 blank squares form the 4th region. Once a board was complete, I then did a count of the 2×2 blocks which were 4 of a single region and threw the board out if there were too many (too boring).

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  • Posted Wed Jan 18, 2017 6:11 pm
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Regarding my solution, oops! In my thinking about LAP I’ve always had the coordinate letters across the top and numbers down the left side, which is not what people did in this thread. Hopefully it will all be clear if you flip around the diagonal.
 
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Solve a Graph Puzzle

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Fifth Grade Math Activities: Solve a Graph Puzzle

What You Need:

  • Graph paper
  • Pencil
  • Ruler

What You Do:

(2,2), (2,4), (3,5), (4,4), (4,2)
  1. Using graph paper, help your child draw a four quadrant plane. Label the x-axis (the horizontal line) and the y-axis (the vertical line). Label the intersecting of both lines 0. Label each point to the right on the x-axis 1 to 7. Label each point to the left on the x-axis -1 to -7. On the y-axis label the points up with positive numbers, 1-7, and then label the points down with negative numbers, -1 to -7. If using the printable Coordinate Plane worksheet, print out and review the x-axis and y-axis
  2. Give your child these ordered pairs and ask him to plot them on the coordinate plane:
(0, -2), (1,-3), (2,-2), (2,-4), (1,-5), (-1,-5), (-2,-4), (-2,-2), (-1,-3)
  1. Remind your child to plot the first number of the ordered pair on the x-axis and the second number of the ordered pair on the y-axis.
  2. Next, have your child take a ruler and connect the 5 points. Ask him what geometric shape he created (a pentagon).
  3. Give your child another set of ordered pairs to graph:
(-2,3), (-3,5), (-4,6), (-5,5), (-6,3), (-4,4), (-2,5), (-6,5)
  1. Have him connect the 9 points and describe the shape he has created (a tulip)
  2. Here’s a challenging puzzle for your child to solve. Have him plot this set of ordered pairs:

This time it is important to connect the points in the order shown. If each ordered pair is graphed accurately and each point is connected in the given order, he will have created a five-pointed star!

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Mystery Picture Graph #1
Worksheet

Mystery Picture Graph #1
Have a little fun with graphing practice in the big city! Math students will love plotting points on this coordinate grid as they see a mystery picture emerge.
5th grade
Math

Worksheet

0.5 Centimeter Graph Paper
Worksheet

0.5 Centimeter Graph Paper
…Provide a helpful math and geometry resource! Use this worksheet for shapes, multiplication, or graphing coordinates….
4th grade
Math

Worksheet

Angle Puzzle
Worksheet

Angle Puzzle
…She’ll use her angle knowledge to steer the ship according to the given directions, using a ruler to draw the direction of each turn….
4th grade
Math

Worksheet

Mystery Picture Graph #2
Worksheet

Mystery Picture Graph #2
They’ll plot these points on a coordinate grid to reveal a picture.
5th grade
Math

Worksheet

Mystery Picture Graph #3
Worksheet

Mystery Picture Graph #3
Make math fun with this coordinate grid activity, where your child will reveal a hidden picture with every point he plots!
5th grade
Math

Worksheet

Grid Drawing
Worksheet

Grid Drawing
The coordinate plane just got a little spookier! Can you graph these coordinates to reveal the hidden picture?
5th grade
Math

Worksheet

Graphing Points on a Coordinate Plane
Exercise

Graphing Points on a Coordinate Plane
Give students a solid foundation in graphing points on coordinate planes with this educational exercise.
5th grade
Math

Exercise

Angle Practice
Worksheet

Angle Practice
This fun puzzle provides lots of angle practice for your child. Can he figure out which way the boat is facing after the last turn?
4th grade
Math

Worksheet

Treasure Map Graphing on a Coordinate Plane
Game

Treasure Map Graphing on a Coordinate Plane
Kids track down buried treasure by plotting coordinates on a map in this fun-filled geometry game.
5th grade
Math

Game

Rescue Mission: Graphing on a Coordinate Plane
Game

Rescue Mission: Graphing on a Coordinate Plane
Kids must plot (x, y) coordinates on a plane to locate an emergency situation in this fun-filled game.
5th grade
Math

Game

Volume and a Building
Lesson plan

Volume and a Building
…Show your students a hundred flat base-ten block, have them turn and tell a neighbor what kind of math problem a math tool like this might help solve….
5th grade
Math

Lesson plan

Cubes and Volume
Worksheet

Cubes and Volume
Then, once they solve the equation have them draw the 3-D objects. The objects will all be rectangular prisms.
5th grade
Math

Worksheet

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