Open Middle®
Directions: Using the digits 1 to 9 at most one time each, place a digit in each box to make a solution that is as close to 100 as possible.
Hint
Hint
How do the upper and lower bounds determine whether the solution is negative or positive?
Answer
Answer
I only know how to figure this out via brute force and would love to learn if anyone has a conceptual approach to this.
I created this spreadsheet and tried all 9 potential exponent values. The bounds and exponent that result in a value as close to 100 as possible are:
upper bound: 8
lower bound: 6
exponent: 2
value: 98.666…
and a close second place is:
upper bound: 2
lower bound: 1
exponent: 9
value: 102.30
Source: Robert Kaplinsky
I created a desmos graph to quickly sift through solutions. I integrated leaving the exponent as a variable, and then used “x” and “y” to be the upper and lower bound and set the whole thing equal to 100. I used the slider to change the exponent “c”, and then found solutions for x and y where x is not equal to y and the points are approximately integral. I came up with exponent 2, upper bound 8, lower bound 6; exponent 3, upper bound 5, lower bound 4; and exponent 9, upper bound 2, lower bound 1. The optimal solution is exponent 2, which is 98.6666666…
Here is the desmos graph I used: https://goo.gl/QEc25d
This is really cool Alison! I had never thought about it visually. I’ll need to think about this more.
One of my students did the integral from 0 to 2.9 for x^5. She wondered if she could use it since the problem didn’t restrict them to integers. 😉
I guess that could work, but by “digit” it was intended to be an integer. Creative thinking though!