The numbers 1-12 are to be placed around a circle, as on a clock, but in any order.
Show that there are three consecutive numbers in the arrangement with a sum of at least 19.
The sum of the 12 numbers is 78.
If we add up all the sums of three adjacent numbers,
we use each number three times, for a total of 234.
If all the sums are to be < 19, then their maximum
total would be 12 * 18 = 216, which is too small.
In fact, there must be a sum of at least 20,
since 12 * 19 = 228, which is still too small.
To avoid a sum of at least 19, the numbers 12, 11, 10, and 9
must all be separated by at least two other numbers.
Otherwise, two of them, plus any of the others totals at least 20.
Let the 12 places be as follows:
_ X _ _ X _ _ 9 _ _ X _
where X indicates 10, 11, and 12 in some order.
Since each of the dashes is part of some 3-element sum
with one of the X's, none of them can accommodate 8,
without creating a sum that is at least 8 + 10 + 1 = 19.