> Look at this:
> 
> while year_one.to_i < year_two.to_i
>   if false
>     if year_one%100 != 0 || year_one%400 == 0
>       total = total + 1
>       year_one = year_one+ 1
>     else
>       year_one = year_one+ 1
>   end
> 
> Also, what value is stored in the variable year_one right before the 
> while loop starts?

I think I've determined my problem; it relates to the variable 
'year_one.' year_one is supposed to be the first year entered when the 
program starts.

Check out my revision, which appears to work:

puts 'Use this calculator to determine the leap years in a specific 
period.'
puts 'Enter the starting point below:'
start = gets.chomp.to_i
puts 'Enter the ending point below:'
ending = gets.chomp.to_i

year = start
total = 0

while year <= ending
  if year%4 == 0
    if year%100 != 0 || year%400 == 0
      puts year
      total = total + 1
    end
  end

  year = year + 1

end

if total == 1
  puts 'There is 1 leap year between '
  puts start.to_s + ' and ' + ending.to_s + '.'
else
  puts 'There are ' + total.to_s + ' leap years between '
  puts start.to_s + ' and ' + ending.to_s + '.'
end
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