Please pass the salt

[BMAD]

I am sitting at a large table for a family reunion. The table carefully places me and my 8 cousins around the table. Given that we like to eat, the host made sure that everyone was placed equidistant from the person to their left and their right. Starting at me and going clockwise there was Eric, Donald, Chris, Becky, Amber, Isaac, Huey, and George. As soon as the food is delivered, Amber asked me to slide her the salt. After she finished with the salt, she passed the salt to Chris and handed pepper to Becky. Prove that the salt traveling from me to Amber is equivalent in distance to the trip the salt and pepper traveled from Amber to Chris and Becky, respectively.

[MuckersMate]

Assumptions

1. Author had the salt to start with, with Amber sitting on the opposite side of the table.

2. Amber had the pepper to start with.

3. Amber handed the pepper directly to Chris, bypassing Becky

Therefore, Salt from Author to Amber = 2 units. (He had to 'slide' it to her because she was too far away to pass it directly).

Salt from Amber to Chris = 1 unit (passed directly)

Pepper from Amber to Becky = 1 unit.

Therefore

2 = 2????

[nakulendu]

Imagine Circular table of radius r. Then cousins will be at vertices of a 9 sided regular polygon inscribed in the circle. The respective distances in question become .

Me to Amber (salt travels) = D1 = 2r COS (10^o)

{if you draw the polygon an put people around it in order then Amber and I will be at the base vertices of an Isoscles triangle which has r as the sides and 10^o being the base angles. The distance Amber-Me = D1 is the base of this triangle}

Amber to Chris (Salt travels) = D2 = 2r COS (50^o) {similarly Amber and Chris are also a the base of another Isoscles triangle with equal sides r and the base angle being 50^{o }}

Amber to Becky (pepper moves) = D3 = [r COS (50o)]/COS (20^o) {try the geometry - Amber - Becky also at base of another isoscles triangle with base angle of 70^o. This time we need to take the cosine of the difference 70^o- 50^o to get the distance D3.}

D1 = 1.969615506024416118733486049179 r

D2 = 1.2855752193730786526452868198145 r

D3 =0.68404028665133746608819922936452 r

D1 = D2 + D3

[tammie]

the table is an equalateral triangle. draw it out with 6 inches on each side. each person sits 2 inches apart. start at the top point with huey then going clockwise every 2 inches.it would be george, the writer, eric(at the right point), donald, chris, becky(at the left point), amber and isaac to the left of the top point(huey). now draw a line from the writer to amber the distance is 4 inches. draw a line from amber to chris distance 2 inches, and finally a line from amber to becky is another 2 inches. 2+2=4. yay!

[k-man]

Nice "out of the box" solution, tammie. Good thinking!

The below picture makes it pretty obvious that the red line segment is equal to the sum of the green line segments. What makes it obvious is the extra line drawn connecting George and Chris. This line divides the BMAD-Amber line into 2 parts and forms 2 equilateral triangles. There are many ways to prove that these triangles are equilateral, but the simplest of them all is drawing another line connecting Donald and Isaac and showing that the triangle formed in the center is equilateral.

[BMAD]

I wish there was a way to designate two correct answers given that i had two people answer in completely different but perfectly logical ways.

the table is an equalateral triangle. draw it out with 6 inches on each side. each person sits 2 inches apart. start at the top point with huey then going clockwise every 2 inches.it would be george, the writer, eric(at the right point), donald, chris, becky(at the left point), amber and isaac to the left of the top point(huey). now draw a line from the writer to amber the distance is 4 inches. draw a line from amber to chris distance 2 inches, and finally a line from amber to becky is another 2 inches. 2+2=4. yay!

Edited February 28, 2013 by BMAD

[BMAD]

Imagine Circular table of radius r. Then cousins will be at vertices of a 9 sided regular polygon inscribed in the circle. The respective distances in question become .

Me to Amber (salt travels) = D1 = 2r COS (10^o)

{if you draw the polygon an put people around it in order then Amber and I will be at the base vertices of an Isoscles triangle which has r as the sides and 10^o being the base angles. The distance Amber-Me = D1 is the base of this triangle}

Amber to Chris (Salt travels) = D2 = 2r COS (50^o) {similarly Amber and Chris are also a the base of another Isoscles triangle with equal sides r and the base angle being 50^{o }}

Amber to Becky (pepper moves) = D3 = [r COS (50o)]/COS (20^o) {try the geometry - Amber - Becky also at base of another isoscles triangle with base angle of 70^o. This time we need to take the cosine of the difference 70^o- 50^o to get the distance D3.}

D1 = 1.969615506024416118733486049179 r

D2 = 1.2855752193730786526452868198145 r

D3 =0.68404028665133746608819922936452 r

D1 = D2 + D3

the table is an equalateral triangle. draw it out with 6 inches on each side. each person sits 2 inches apart. start at the top point with huey then going clockwise every 2 inches.it would be george, the writer, eric(at the right point), donald, chris, becky(at the left point), amber and isaac to the left of the top point(huey). now draw a line from the writer to amber the distance is 4 inches. draw a line from amber to chris distance 2 inches, and finally a line from amber to becky is another 2 inches. 2+2=4. yay!

was just taught how to mark two best answers!

Edited February 28, 2013 by BMAD