Open Middle®
Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two points that are equidistant from (4,-1).
Hint
Which methods are available to determine answers to this problem? What shape is defined by all of the points that are equidistant from (4, -1)?
Answer
There are many correct answers to this problem including: (2, 4) and (9, 1) as well as (2, 3), and (8, 1)
Source: Bryan Anderson
We worked this problem out and only found 4 possible answers. Do you think there are more than that?
Our other answers were (1,4), (9,2) and (1,3), (8,2).
Thanks!
there are many more, if you think in terms of slope, any that create perpendicular lines will work; all the points to the left or right two and up one will be the same as going left or right one, then up two. etc
If remove the limits of only being able to use the digits 1 through 9 at most once, then there are infinite possibilities, but with that limit it place I think Andrew is right. I can only find the two pairs listed above. Any other pairs of points will require repeating a digit or using 0, negatives, or fractions.
I just gave this to my students and here are some of the answers that I have verified as correct. There may be more if you count the triangles on the grid, like I did. I imagine there are several more that could be correct.
(1,3) & (8,2)
(2,3) & (8,1)
(8,1) & (6,3)
(6,2) & (7,1)
(3,7) & (8,6)- this one was very creative! Distance is sqrt(65)
(1,4) & (9,2)
Here are all solutions to this problem, I determined this by using pythagorean theorem
Moves: 2-3 distance = sqrt 13 (7,1) (6,2)
Moves: 2-4 d = sqrt 20 (8,1) (6,3)or(2,3)
Moves: 2-5 d = sqrt 29 (9,1) (6,4)or(2,4)
Moves: 3-4 d = sqrt 25 (8,2) (1,3)or(7,3)
Moves 3-5 d = sqrt 34 (9,2) (7,4)or(1,4)
Moves 4-5 d = sqrt 41 (9,3) (8,4)
Additionally: Moves 4-7 (8,6) and 1-8 (3,7) both have a distance of sqrt 65
I believe this is all possible solutions, but I am continuing to test possible solutions