More handshakes

thickhead · 2016-05-02 02:12:24

Why phrontister's excel table does not have row for A ? and why in the main diagonal "x" is not put?

thickhead · 2016-05-02 02:20:00

A will have 2+2+2 ... 31 times=62
B will have 30*2+3=63
C will have 29*2+2*3=64
"e" will have 2*2+29*3=91
"f" will have 2+30*3=92

phrontister · 2016-05-02 02:56:38

Ah...yes! Exactly, thickhead.

X = 3.

And thanks for spotting the flaws in my table.

thickhead wrote:

"e" will have 2*2+29*3=91
"f" will have 2+30*3=92

I've got the following for those:
e = 2 + 30*3 = 92;
f = 31*3 = 93

Because f is the 31st term from A, and A = 62, then f = 62 + 31 = 93...and e is 1 less.

Last edited by phrontister (2016-05-02 03:16:05)

thickhead · 2016-05-02 04:37:26

In your first line under "f" the term should be 2;Please see the symmetrical 1st point on the last line for "f" under "A" It is 2. The sheet is symmetrical albout the main diagonal.The proposal is "A" will have 2 handshakes with all others including "f".

anna_gg · 2016-05-02 05:05:46

Yes but then the total will be 92 instead of 93, which will be the same with the previous one!

thickhead wrote:

In your first line under "f" the term should be 2;Please see the symmetrical 1st point on the last line for "f" under "A" It is 2. The sheet is symmetrical albout the main diagonal.The proposal is "A" will have 2 handshakes with all others including "f".

anna_gg · 2016-05-02 09:02:27

And also last A17 with A16 is 3, while A16 with A17 is 2.

anna_gg wrote:

Yes but then the total will be 92 instead of 93, which will be the same with the previous one!
thickhead wrote:
In your first line under "f" the term should be 2;Please see the symmetrical 1st point on the last line for "f" under "A" It is 2. The sheet is symmetrical albout the main diagonal.The proposal is "A" will have 2 handshakes with all others including "f".

phrontister · 2016-05-02 09:05:28

The table is perfectly symmetrical about the 'x' diagonal.

eg,
Row "A" has 1 @ "x", then 30 @ "2", then 1 @ "3", and row "f" has 1 @ "2", then 30 @ "3", then 1 @ "x";
Column "A" has 1 @ "x", then 31 @ "2", and column "f" has 31 @ "3" then 1 @ "x".

Also, starting with column A = 62, the A.P. of +1 requires the following handshake totals:
A=62, B=63, C=64...d=91, e=92, f=93.

And the total for column "f" can only be 93 by 31*3 and 1 @ "x".

Also, a Countif search of the table finds 32 @ "x", 496 @ "2" and 496 @ "3".

Last edited by phrontister (2016-05-02 09:21:02)

anna_gg · 2016-05-02 09:20:59

I don't get it. If Row "A" has 1 @ "x", 30 @ "2" and 1 @ "3", then shouldn't Column A have also the same? How come it has 31 @ "2" (instead of 30) and none @ "3"?
If the intersection of, say, A2 with A31 is 3, then also A31 with A2 should be 3 (not 2), right??

phrontister wrote:

The table is perfectly symmetrical about the 'x' diagonal.
eg,
Row "A" has 1 @ "x", then 30 @ "2", then 1 @ "3", and row "f" has 1 @ "2", then 30 @ "3", then 1 @ "x";
Column "A" has 1 @ "x", then 31 @ "2", and column "f" has 31 @ "3" then 1 @ "x".
Also, starting with column A = 62, the A.P. of +1 requires the following handshake totals:
A=62, B=63, C=64...d=91, e=92, f=93.

phrontister · 2016-05-02 09:43:18

Hi Anna;

You should compare columns with columns and rows with rows, not columns with rows. The 'x' diagonal only goes in one direction (NW to SE), which throws the expected balance out, but there is another way of looking at this that shows we have the balance we need.

Here are two coloured images from my spreadsheet to help demonstrate. All green cells contain "2" and all blue cells "3".

In this version, if you draw a line through the centre to two cells opposite to, and equidistant from, each other, you'll find that one of them will contain a "2" and the other a "3". Always. In the following examples of this, the first letter denotes the row and the second the column.

Ae ("2") is opposite fB ("3").
OR ("3") is opposite RO ("2")

And so forth, and so fifth...etc.

There are an equal number of each colour/number, distributed equally in a balanced pattern about the 'x' diagonal.

This version comprises two diagonally-cut halves (with the NE sector rotated 180°) alongside each other for comparison. You'll see that all "2"s and "3"s in the right-way-up image align exactly with their respective "3"s and "2"s counterparts in the 180°-rotated image. That pattern applies throughout, which could not happen without 100% balance.

Brimming with balance!

Just had a thought and checked my images. There's a critical error! Grumble! Back to the drawing board!! Maybe you're right. Have to go out now for a few hours, and will look at this again this evening.

Last edited by phrontister (2017-02-23 11:36:55)

thickhead · 2016-05-02 16:00:20

Last night I had an idea of what happened. We started with A having 2 handshakes with all.with b all but the last "f" had 2 handshakes and with "f" 3 handshakes. So the wave of "3" advanced leftwards on a lesser than longest backward diagonal. in the chess board terminology this diagonal had the same color as the main right diagonal containing "x" value. So the two diagonals crossed and the 16th man(Q)and 17th man(R) had to have same no. of handshakes. If we had started with 3 on the right top corner the two longest diagonals do not intersect and the problem does not arise. I was to post the same today morning but you were one step ahead of me.We can repair the previous set up by using 4 handshakes with "f" for everybody from "Q" to "e".

phrontister · 2016-05-02 16:09:23

Sorry, thickhead, but I have to rush off now for a few hours and can't spend time on your post until this evening.

I'd logged on again to edit my post #34, which I finished just a few minutes ago.

Catch you later...

Quick edit before I go: Just had a thought and checked my images. There's a critical error! Grumble! Back to the drawing board!!

Last edited by phrontister (2016-05-02 16:18:32)

thickhead · 2016-05-02 17:22:23

When we have finished with "P" and are going to "Q" we find another 3 can not be added as the diagonal joining (f,B) to (B,f) falls into a ditch ("x" of the main diagonal )and the total for Q remains as that of "P" viz. 77. this difficulty may be overcome by revising Q's handshake with "f" to 4 instead of 3.That should be continued to all the persons from "R" to "e".

Last edited by thickhead (2016-05-02 18:02:15)

anna_gg · 2016-05-02 17:58:06

Last one for f with f should be x, not 4.

thickhead wrote:

When we have finished with "P" and are going to "Q" we find another 3 can not be added as the diagonal joining (f,B) to (B,f) falls into a ditch ("x" of the main diagonal )and the total for Q remains as that of "P" viz. 77. this difficulty may be overcome by revising Q's handshake with "f" to 4 instead of 3.That should be continued to all the persons from "R" to "e".

thickhead · 2016-05-02 18:03:55

Thanks Anna, correction made.I was doing routine copying from one cell.Hence the mistake.

phrontister · 2016-05-02 18:06:32

Raining outside...can't dig holes in the mud!

This is the best I've got so far:

X = 4.

Last edited by phrontister (2017-02-23 11:29:48)

thickhead · 2016-05-02 18:22:46

In this I implemented value 3 from the top right to bottom left diagonal. Here also totals for P and Q are same value being 78. I had thought in the previous case it was because of actual crossing of diagonals. But it is not so.So it becomes necessary to go for 4 handshakes. it means minimum number of X asked in the puzzle is 4.

phrontister · 2016-05-02 22:18:11

Hi;

I found another answer, but still for X = 4.

It uses 12 less "4"s than in my previous solution.

I tried for X = 3 again, but can't find a solution for it.

Last edited by phrontister (2017-02-23 11:31:05)

anna_gg · 2016-05-02 22:23:00

Now it makes more sense!! Despite the rain

samuel.bradley.99 · 2016-05-02 22:26:39

I agree with your solution guys (not, of course, that I could have figured it out myself!!).
It is close to what I tried to do via calculations and without the use of excel (adding all possible different numbers starting from 62 and dividing by the number of pairs).

Congrats to both of you!

phrontister · 2016-05-02 22:39:13

Definitely a team effort!

I'd still be poking away at it in the dark if it wasn't for the insight of the other team members.

anna_gg · 2016-05-03 00:04:54

Hey guys you are both so clever!

thickhead · 2016-05-03 00:34:46

phrontister wrote:

Hi;
I found another answer, but still for X = 4.
It uses 12 less "4"s than in my previous solution.
I tried for X = 3 again, but can't find a solution for it.

Now it is more picturesque than before.

phrontister · 2016-05-03 10:06:46

I wonder if the minimum number of "4"s is 16. But I won't look any further.

thickhead wrote:

Now it is more picturesque than before.

I used Excel's 'Conditional Formatting' on the whole 32 x 32 grid to automatically apply the different colours according to which number I entered into a cell. Very quick and easy. It helps me keep track visually of what I'm doing, and makes the end result easy to understand.

The colouring also helped confirm that the last two grids were error-free...which I hadn't done properly at first, hence my earlier errors.

Well, the rain has finally stopped, but the builders next door have started...and so has their noise! So I may as well go outside and see if my noise can beat theirs: my 9" angle grinder and petrol-driven posthole digger versus their generator, pneumatic nailing guns and rap 'music'! It's war!!

Last edited by phrontister (2016-05-03 13:18:17)

anna_gg · 2016-05-03 23:30:10

Earplugs and lots of puzzles to keep busy!!!

thickhead · 2016-05-04 01:46:04

I am jealous of phrontister living in a zone currently raining. We in India are deprived of rains and suffering from scarcity of water.

Math Is Fun Forum

#26 2016-05-02 02:12:24

Re: More handshakes

#27 2016-05-02 02:20:00

Re: More handshakes

#28 2016-05-02 02:56:38

Re: More handshakes

#29 2016-05-02 04:37:26

Re: More handshakes

#30 2016-05-02 05:05:46

Re: More handshakes

#31 2016-05-02 09:02:27

Re: More handshakes

#32 2016-05-02 09:05:28

Re: More handshakes

#33 2016-05-02 09:20:59

Re: More handshakes

#34 2016-05-02 09:43:18

Re: More handshakes

#35 2016-05-02 16:00:20

Re: More handshakes

#36 2016-05-02 16:09:23

Re: More handshakes

#37 2016-05-02 17:22:23

Re: More handshakes

#38 2016-05-02 17:58:06

Re: More handshakes

#39 2016-05-02 18:03:55

Re: More handshakes

#40 2016-05-02 18:06:32

Re: More handshakes

#41 2016-05-02 18:22:46

Re: More handshakes

#42 2016-05-02 22:18:11

Re: More handshakes

#43 2016-05-02 22:23:00

Re: More handshakes

#44 2016-05-02 22:26:39

Re: More handshakes

#45 2016-05-02 22:39:13

Re: More handshakes

#46 2016-05-03 00:04:54

Re: More handshakes

#47 2016-05-03 00:34:46

Re: More handshakes

#48 2016-05-03 10:06:46

Re: More handshakes

#49 2016-05-03 23:30:10

Re: More handshakes

#50 2016-05-04 01:46:04

Re: More handshakes

Board footer