Positive Differences

Can you fill the circles with the numbers 1 to 6?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

In the diagram below the number in each circle is the (positive) difference of the numbers in the two circles connected to it below. 

Is it possible to to fill in the circles with the numbers $1$ to $6$, using each number exactly once?



Image
Positive Differences


Can you show that:

  • $6$ must be in the bottom row?
  • $5$ cannot be at the top?
  • $4$ cannot be at the top?

Can you find all of the possible solutions?

 

Extension

Can you fill these circles with the numbers $1$ to $10?$

Can you find all of the possible solutions?

Image
Positive Differences


With thanks to Don Steward, whose ideas formed the basis of this problem.