Find all distinct pentagons you can draw on a 3x3 9-pin square pinboard

[MattDunbar]

hi all

on a 3x3 9-pin square pinboard you can construct

8 distinct triangles

16 distinct quadrilaterals

but how many pentagons?

the attachment jpeg shows 23 I have found (the data below each is perimeter & area)

I feel there must be a missing 24th pentagon - anyone ale to construct it

your answer please as a diagram or perhaps give the 5 coordinate pairs e.g. (1, 0) (0, 1) (1, 2) (1, 1) (2, 0)

TIA. Matt

 

[Deleted member 4993]

You missed the easiest one:
1* 2*
4* 5* 6*
7* 8* 9*

1-4-7-8-9-6-5-2-1

perimeter = 8, area=3

Curious: any reason for the random numbering of your pentagons?

EDIT:
Are you saying maximum is 24? If so, here's a 25th(!):
1 2 3
4 5 6
7 8 9

1-8-5-6-3-2-1

I believe the first one is there - located in 3rd row 4 th column.

Also the pentagon at 2 nd row 3 rd column is rotated version of your 25 th one.

 

[MattDunbar]

You missed the easiest one:
1* 2*
4* 5* 6*
7* 8* 9*

1-4-7-8-9-6-5-2-1

perimeter = 8, area=3

Curious: any reason for the random numbering of your pentagons?

EDIT:
Are you saying maximum is 24? If so, here's a 25th(!):
1 2 3
4 5 6
7 8 9

1-8-5-6-3-2-1

sorry Denis your first suggestion is a hexagon - 6 sides -I need a new pentagon please which has 5 sides only

and your second suggestion is already represented as pentagon 8 (my numbering does from easiest find to hardest)

any other ideas - if I've misunderstood your suggestion please clarify with (x, y) coordinates please

TIA. Matt

 

[MattDunbar]

I believe the first one is there - located in 3rd row 4 th column.

Also the pentagon at 2 nd row 3 rd column is rotated version of your 25 th one.

yes - you're right about Denis's 2 suggestions which have already been explored

any ideas yourself to get the 24th missing pentagon please ??

or do you think 23 is the limit??

23 is odd, prime and breaks the 8, 16, 24, pattern i thought would appear

although 23 is a very significant number in many respects

TIA. Matt

 

[MattDunbar]

OK...I'm going to the corner....for 7 * 2 = 14 minutes :sad:

sometimes grade school maths is harder than degree level

I widen my challenge to those of you with maths degrees to solve this -

the first reply to get the 24th pentagon is worthy of a place on the first INTERSTELLAR space ship

 

[MattDunbar]

From a site after googling "pentagons in a 9 dot grid":
A warning
Students can all too easily become drawn into looking for numerical patterns within geometric situations. So, for example there is a danger that because there are 8 different triangles and 16 different quadrilaterals that can be made on a 9-pin grid, we might generalise there are 24 pentagons. This is not the case. However, searching out and classifying the different pentagons using similar properties to the way we define quadrilaterals (parallel sides, right angles, equal sides)would be a significant undertaking.

Positions
.1 .2 .3
.4 .5 .6
.7 .8 .9

After examining the 23 you provided, I noticed that 21 of them can start (if repositioned)
at position 1; then by keeping the number (representing the positions making up the 5
sides) in ascending order (your assigned number in brackets):
1 (22) : 1-2-5-3-8-1
2 (14) : 1-2-5-6-8-1
3 (12) : 1-2-5-9-4-1
4 (18) : 1-2-6-5-8-1
5 (16) : 1-2-6-7-5-1
6 (02) : 1-2-6-8-4-1
7 (21) : 1-2-6-8-5-1
8 (04) : 1-2-6-9-4-1
9 (13) : 1-2-9-4-5-1
10(17): 1-2-9-5-8-1
11(15): 1-2-9-8-4-1
12(11): 1-3-5-6-7-1
13(19): 1-3-5-6-8-1
14(09): 1-3-5-8-4-1
15(10): 1-3-5-9-4-1
16(23): 1-3-5-9-8-1
17(07): 1-3-6-5-7-1
18(01): 1-3-6-8-4-1
19(03): 1-3-6-8-7-1
20(08): 1-3-8-5-4-1
21(05): 1-3-9-5-7-1
22(06): 2-6-8-5-4-2
23(20): 2-9-5-8-4-2

great work Denis -I couldn't locate this web page but your search also turned up the following from a Key Maths UK GCSE book...

" It is worth noting there are not 32 pentagons as you might think. There are in fact 31."

This is suggested as supplementary work so even the teacher's guide for this text book will not list the shapes or provide a diagram of them

However it is one less than the predicted value 31 = 32 - 1

Shall we Great Mathematicians conclude this is an error ?

Is the answer 23 = 24 - 1

and is not a predictable value in this case

There is no way you and I have missed a further 8 pentagons?!

unless these missing 8 break the rules and are compound shapes with 2 discrete enclosed areas within them

TIA

Matt

 

[MattDunbar]

thanks all