Records Challenge 23

G. Carelli
Italy

 1 Is this pentomino flippable? I'm not sure that "flippable" it's an English word, I hope you can undestand the meaning, if not let me know. 2 It's possible to cover this pentomino with a domino and a I trimino? Image 3 It's possible to cover this pentomino with a T-tetromino and a single square? Image 4 It's possible to move a single square to reach the W-pentomino? Image

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N N 2 N Y Y Y Y Y N Y N N N N 3 Y N N N Y Y N N N Y Y N 4 Y N N Y Y N N Y Y N N Y

Jeroen De Vos
Belgium

 1 Is the piece symmetric to an axis? 2 Is the number of 90° angles 6 or 7? Image 3 Is it the net of an open box and  doesn't contains a S-tetromino? Image 4 Can you construct the pentomino out of a  I-tromino and 2 monomino's, but the pentomino mustn't contain a I-tetromino or a  T-tetromino? Image

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N N 2 Y N N Y N Y Y N Y N Y Y 3 N N Y N N Y N N N Y Y Y 4 N N N Y N N Y Y N N N Y

Peter Esser
Germany

 1 Is the piece a combination of the domino and the straight tromino? 2 Is the piece symmetric to an axis? 3 Has the piece exactly two ends i.e. two squares with only one connection? Image 4 Is the number of inner corners (270° angles) odd? Image

 F I L N P T U V W X Y Z 1 N Y Y Y Y Y N Y N N N N 2 N Y N N N Y Y Y Y Y N N 3 N Y Y Y N N Y Y Y N N Y 4 Y N Y N Y N N Y Y N N N

B. Henderson
USA

 1 Any Symmetry (across line and/or point) ? 2 Contains L tetromino but not I tetromino? Image 3 Has some ends pointed 90 degees apart? Image 4 Has some ends pointed 180 degrees apart? Image 5 Line symmetry? 6 Contains L tetromino but not N tetromino?

Each scheme uses the relative directions of the pentomino ends in 2 of the 4 classification criteria, so I should explain that these "ends" and their "directions" are the same as in Martin Watson's recent Pentomino Pipes challenge.

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N Y 2 Y N N Y Y Y Y Y N N N Y 3 Y N Y N N Y N Y Y Y Y N 4 Y Y N N Y Y N N N Y Y Y

 F I L N P T U V W X Y Z 3 Y N Y N N Y N Y Y Y Y N 4 Y Y N N Y Y N N N Y Y Y 5 N Y N N N Y Y Y Y Y N N 6 N N Y N N Y Y Y N N Y Y

Bob Henderson mails us:"I found other classifications that depend on the letter of the alphabet each pentomino resembles, but I think that the geometric categories are less arbitrary."
He had the same idea as Martin Watson.

Tom Jolly
USA

 1 Does the piece have line symmetry? 2 Can it be placed flat on a table so that exactly 3 cubes touch the table? Image 3 Can you draw a line on it connecting all the cubes without backtracking? Image 4 Can you add or subtract 1 cube to turn the piece into a rectangle? Image

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N N 2 N N N Y Y Y Y Y N N N N 3 N Y Y Y Y N Y Y Y N N Y 4 N Y Y N Y N Y N N N Y N

E. Künzell
Germany

 1 Can you construct the pentomino out of a domino and the triomino in the shape of a rectangle (3 sqares in a row)? 2 Is it possible to construct the 3x3 sqare and the tenth sqare added at one corner with the given one and another second pentomino? ( The task is similar to the one in the 4th question, but the shape is not like a house, but rather like a lorry)? 3 Double the size of the pentomino by using 4 different pentominoes. Are there more than 2 different solutions? Image 4 Can this pentomino together with another pentomino make the following decamino? Image

 F I L N P T U V W X Y Z 1 N Y Y Y Y Y N Y N N N N 2 Y N Y N Y Y Y Y N N Y Y 3 N N Y Y Y N Y N Y N N Y 4 Y N N Y Y N Y Y N Y N N

Helmut Postl
Austrich

 1 Can this pentomino together with another pentomino be used to cover the following shape? 2 Does the pentomino fit into a 2x5-rectangle? Image 3 Does the pentomino contain a T-tetromino? Image 4 : Does the pentomino have exactly 8 edges? Image

 F I L N P T U V W X Y Z 1 Y N Y N Y Y Y Y N N Y Y 2 N Y Y Y Y N Y N N N Y N 3 Y N N N Y Y N N N Y Y N 4 N N N Y N Y Y N N N Y Y

Jaap
Scherphuis

 1 Does it have line symmetry? 2 Does it fit inside this figure (a 3x3 square with a 1x2 corner missing)?      x      xxx      xxx Image 3 Count the number of squares with exactly two neighbouring squares, on opposing sides. Is this an odd number? Image 4 Count the number of squares with exactly two neighbouring squares, on non-opposing sides. Are there at least two such squares? Image

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N N 2 Y N N N Y Y Y Y Y N N Y 3 N Y N Y N Y Y N N N Y Y 4 N N N Y Y N Y N Y N N Y

Peter Sipos (Hongarije)

 1 Does it have line symmetry? 2 Is the adjacency graph of the constituting squares a path (node degrees < 3)? Image 3 Is the area of the bounding rectangle less than 9? Image 4 Can the following decamino be solved using this pentomino?

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N N 2 N Y Y Y N N Y Y Y N N Y 3 N Y Y Y Y N Y N N N Y N 4 Y N N Y Y N Y Y N Y N N

v. d. Wetering

 1 Do you have line symmetry? 2 Do you fit inside a 3x3 square, touching all sides? Image 3 Are the numbers of your squares' individual outer edges - sorted - 2-2-2-3-3? Image 4 Do two copies fit inside a 2x5 rectangle and/or inside the figure shown below?       X       XXXX       XXXX              X

 F I L N P T U V W X Y Z 1 N Y N N N Y Y Y Y Y N N 2 Y N N N N Y N Y Y Y N Y 3 N Y Y Y N N Y Y Y N N Y 4 N Y Y N Y Y N Y N N N N

 Berend Jan  v. d. Zwaag
 1 Does it contain two non-4-touching dominoes? (Touching at the corners is allowed.) Image 2 Does it fit together with a heptomino in a 3x4 rectangle? Image 3 Does it fit together with two copies of another pentomino in a 3x5 rectangle? Image 4 Does it have line-symmetry? 5 Can it be constructed using a domino and a straight tromino? 6 Does it fit inside a 2x5 rectangle? 7 Does it fit inside a 3x3 square? 8 Does it fit together with a tromino and a tetromino in a 3x4 rectangle? 9 Is its bounding rectangle a square? 10 Does it contain two non-8-touching dominoes? (Touching at the corners is not allowed.) 11 Does it fit together with a domino and two monominoes in a 3x3 square? 12 Does it fit twice in a 3x4 rectangle? 13 Does it fit together with a tromino and a domino in a 2x5 rectangle? 14 Can two copies construct a decamino with a fully enclosed 1x1 hole? 15 Can two copies construct a point-symmetric decamino with a fully enclosed 1x1 hole?

 F I L N P T U V W X Y Z 1 N Y Y Y N N Y Y Y N N Y 2 N N Y N Y Y Y Y N N Y N 3 N Y Y N N Y N Y N Y Y Y 4 N Y N N N Y Y Y Y Y N N N Y Y Y Y Y N Y N N N N 6 N Y Y Y Y N Y N N N Y N 7 Y N N N Y Y Y Y Y Y N Y 8 N N Y N Y Y Y Y Y N Y N 9 Y N N N N Y N Y Y Y N Y 10 N Y Y N N N Y N N N N Y 11 Y N N N Y Y Y Y Y N N Y 12 N N Y Y Y N Y Y Y N Y N 13 N Y Y Y Y N N N N N Y N 14 Y N Y Y N Y Y N N N Y Y 15 Y N Y Y N Y N N N N Y Y

With these 15 questions there are 1365 different possible combinations of 4 questions, of which only 29 different combinations each uniquely determine the pentominoes:
 1-2-3-4 1-2-3-6 1-2-3-7 1-2-3-9 1-2-3-11 1-2-3-13 1-2-4-5 1-2-5-14 1-2-5-15 1-2-6-15 1-2-7-14 1-2-9-15 1-2-11-14 1-2-13-14 1-3-6-8 1-3-6-15 1-3-7-14 1-3-8-9 1-3-9-15 1-3-11-14 1-3-13-14 1-5-8-14 3-5-6-14 3-5-6-15 3-5-9-14 3-5-9-15 3-5-10-14 3-5-10-15 3-5-12-14

J. Viljoen mails us:"Herewith my lazy solution.Yes, I know it is cheating. Or is it? It seems to meet the specifications of the challenge, at least the way I read them."
Alphabetise the pentominoes as is the custom: F I L N P T U V W X Y Z. Now, using this alphabetical order, number them from 1 to 12. F will therefore be 1, and Z will be 12. Turn this number into a 4-bit binary number.

J. Viljoen
South-Africa

 1 Is the first bit set (ie equal to 1)? 2 Is the second bit set (ie equal to 1)? 3 Is the third bit set (ie equal to 1)? 4 Is the fourth bit set (ie equal to 1)?

 F :0001 I:0010 L:0011 N:0100 P:0101 T:0110 U:0111 V:1000 W:1001 X:1010 Y:1011 Z:1100 1 Y N Y N Y N Y N Y N Y N 2 N Y Y N N Y Y N N Y Y N 3 N N N Y Y Y Y N N N N Y 4 N N N N N N N Y Y Y Y Y

M.H. Watson
England

 1 Is this pentomino V,W,X,Y or Z? 2 Is this pentomino N,P,T,U or Z? 3 Is this pentomino I,L,T,U,X or Y? 4 Is this pentomino F,L,P,U,W or Y?

 F I L N P T U V W X Y Z 1 N N N N N N N Y Y Y Y Y 2 N N N Y Y Y Y N N N N Y 3 N Y Y N N Y Y N N Y Y N 4 Y N Y N Y N Y N Y N Y N

Martin mails us: "This probably isn’t the kind of answer you wanted, but I like it. I am sure your class will enjoy it!!
This is effectively allocating each pentomino a binary number from F=0001 upto Z=1100. Each ‘1’ corresponds with a YES.
Every pentomino has a different answer. This would still work if there were 16 pentominoes, from 0000 to 1111."
We have nothing to add to this.

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