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Even an Odd Problem: Just Playing Around....

While walking the dogs one morning, a small little problem came to mind. The problem perhaps is quite trivial, but it is a nice play on vocabulary words in mathematics.

The Original Question: Is it possible that the sum of an even number of odd numbers can equal the sum of an odd number of even numbers?

Too easy....so lets try a small variation....

The Revised Question: Is it possible that the sum of an even number of consecutive odd numbers can equal the sum of an odd number of consecutive even numbers?

Play with this idea until you have a solution....then try these additional restrictions....

  • What is smallest number of odds and evens that will make this question solveable?
  • Given any finite sequence of an even number of consecutive odd numbers, can one always find its "paired" finite sequence of an odd number of even numbers?
  • Is it possible that that both sequences have an infinite number of numbers in them?
  • And finally, is it possible that one sequence has an infinite number of numbers and the other a finite number of numbers?
Have fun exploring......my dogs did!

 

Source: To quote E.A.: "My own wee brain!"


Hint: Two things to remember:

  1. Odd and even numbers can be negative
  2. Zero is an even number

 


Solution Commentary: Some not-so random musing:

  • As one sequence pair, -2 + 0 + 2 = -1 + 1....but there is an even smaller sequence pair!
  • Notice that 3 + 5 + 7 + 9 = 6 + 8 + 10....which implies that 5 + 7 + 9 + 11 + 13 = sum of what sequence? Look for a pattern?...so that if given the odd sequence n + (n+2) +...+ (n+2m), what would be the associated even sequence of odd numbers?