Problem number 14 states, The first 1,000 positive integers are written in a row in some order, and all sums of any three adjacent numbers are calculated. What is the greatest number of odd sums that can be obtained? So we'll have 1,000 numbers, and we want to maximize the number of odd sums between three adjacent numbers, so we can say we want to maximize the number of odd sums of ax plus ax plus 1 plus ax plus 2. Ax must be between 1 and 998 inclusive, so we have a maximum of 998 possible sums. Between the numbers 1 and 1,000, we have 500 even numbers and 500 odd numbers. So how exactly do we get an odd sum? Well we get an odd sum if we add an odd number of odd numbers. So for example, 1 plus 3 is 4, which is adding an even number, 2, of odd numbers to the sum, so we get an even value of 4. If we add another odd number, then we'll be adding an odd number, 3, of odd numbers, so the sum is odd, 9. If we were to add an even number, it wouldn't matter, as we were still adding an odd number of odd numbers. So one pattern that we could possibly think of is if we just had three odds in a row. No matter how you shift the reference frame, you will always add three odds together, which will be an odd number. However, we don't have 1,000 odds, so we need to find another pattern. And that can happen with one odd and two evens in a repeating pattern. Again, no matter how we shift the reference frame, there will always be one odd number, and thus the sum will always be odd. So we have 500 odd numbers and 500 even numbers. Let's figure out how we will put these two patterns together. So the second pattern requires twice as many evens as odds, so let's use all of our evens there and put all the odds for the remaining numbers. So we have 500 evens, which means that the second pattern of one odd and two evens can continue for 750 numbers. And of these numbers, 250 of them will be odd, which means the triple odd pattern will continue for 250 numbers. So this is how the pattern would look like. And we can notice that at the transition, one of the sums is going to end up being even. So in total, we had 998 sums. And the best we could do is make 997 of them odd, because the one in the transition ends up with an even number. So the question asked us, what is the greatest number of odd sums that can be obtained? The answer is 997, letter A.

odd sums

integer arrangement

odd and even numbers

pattern sequence

maximum odd sums