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Where? And Up or Down?

Some problems prove more interesting to students, and I am rarely sure why. The following problem is one that students (and math teachers) have enjoyed exploring.

Assume the pattern continues...
  1. Where would 125 be located?
  2. Is 125 on a triangle that points up or down?
  3. Where would 457 be located?
  4. Is 457 on a triangle that points up or down?
One nice observation is that numerous students, without being prompted, have solved this extension of the problem:
  • Where would the positive integer N be located?
  • Is the positive integer N on a triangle that points up or down?
Answers to these extensions requires multiple cases, and considerable thought!


Source: Wish I could remember so I could thank the person!

Hint: Write out some more cases....Look for patterns.

Do not waste your time (and thinking ability) by extending the pattern visually to 125, or evebn worse, 457!


Solution Commentary: Share your answers, observations, and conclusions with others exploring the problem. You can be referees for each other.