Open Middle®
Directions: Using the digits 1 to 9 at most one time each, place a digit in each box to make a true pattern where the pattern increases by the smallest amount possible.
Hint
How can we tell whether the digits we place in the boxes actually make a pattern? How can we tell if the pattern increases each time by the digit we choose? How can we tell what would be a bad digit to choose for how much it increases by?
Answer
The pattern that increases by the smallest amount possible is 9, 16, 23 which increases by 7.
Source: Robert Kaplinsky
Suggested by my student, Nicole: 8, 14, 20
Ooops! We did not read that the digit, 0, was not permitted.
Zero would have been helpful!
6, 09, 12 … 3
The 7 ways to fill the boxes according to the rules are:
5, 14, 23 increases by 9
6, 15, 24 increases by 9
7, 15, 23 increases by 8
7, 16, 25 increases by 9
8, 17, 26 increases by 9
9, 16, 23 increases by 7
9, 17, 25 increases by 8
The one where the pattern increases by the smallest amount possible therefore is indeed 9, 16, 23 increases by 7. If we allowed the zero (except in the the tenth’s place), we’d only add two solutions:
6, 13, 20 increases by 7 and
8, 14, 20 increases by 6, which makes this one the one where the pattern increases by the smallest amount possible.
9, 16, 23, increases by 7