Solution

Connecting Averages Arvee K

Problem / game
Connecting averages Back to problem
First name
Arvee K
School
Dr Challoner's High School
School's Twitter handle
@ChallonersHigh
Country
England
Age
12
File
My question for NRICH_1.jpg

My question is in a file that I have sent below, it is labeled as 'My question for NRICH'. This is my explanation:

This network is a discrete Dirichlet problem where each square node is the precise arithmetic mean of its four cardinal neighbors. Because the variables are interdependent, we must solve them simultaneously using a system of linear equations.

The Algebraic proof:

  • Node x: x = (10 + 10 - 4 + y) / 4 --> 4x = 16 + y --> y = 4x - 16
  • Node z: z = (-5 - 2 - 1 + y) / 4 --> 4z = -8 + y --> y = 4z + 8
  • Node y: y = (-1 - 1 + x + z) / 4 --> 4y = -2 + x + z

4x - 16 = 4z + 8 --> 4x - 4z = 24 --> z = x - 6

4(4x - 16) = -2 + x + (x - 6)
16x - 64 = 2x - 8
14x = 56
x = 4

  • y = 4(4) - 16 = 0
  • z = 4 - 6 = -2

Check:

  • (10 + 10 - 4 + 0) / 4 = 4
  • (-1 - 1 + 4 - 2) / 4 = 0
  • (-5 - 2 - 1 + 0) / 4 = -2

 

Final Solution:x=4, y = 0, z = -2