The Walker fanily likes to puzzle people with pictures of themselves standing in five holes they dug in their yard. Here are two such pictures, both taken from the same direction. Each hole has a different depth, and each family member is a different height. By studying the two pictures, can you determine how Mom, Dad, Bro, Sis, and Pup can choose holes that will make them all appear to be the same height?
Solution to the Problem:
From left to right, the arrangement in which everyone will appear the same height is:
Pup, Bro (striped shirt), Dad (red shirt), Mom (blue top), Sis (orange top).
Imagine that the holes are numbered 1 through 5 from left to right. The family members
are shown as having segments that represent units of height. In the first picture, 8 segments of
Mom are showing; let's assume she is 9 units tall, in which case hole #1 is 1 unit deep. (Only the
relative lengths and depths matter. The solution will be identical if we assume that Mom is 12 units
tall and hole #1 is 4 units deep, etc.) Since Mom projects 8 units above hole #1 and only 4 units
above hole #3 in the second picture, hole #3 must be 1 + 4 = 5 units deep. Sis stands 2 units above
hole #3 and 5 units above hole #5, so hole #5 is 5 -3 = 2 units deep. Dad sticks out 8 units above
hole #5 and 6 units above hole #4, so hole #4 is 2 + 2 = 4 units deep. Pup stands 2 units above hole
#4 and 3 units above hole #2, so hole #2 is 4- 1 = 3 units deep. So from left to right the holes are
1, 3, 5, 4, and 2 units deep.
The heights of the family members are their visible lengths plus the
depths of the holes.
Mom 9 units tall, Bro 8 units, Sis is 7 units, Pup is 6 units,
and Dad is 10 units.
To make everyone appear the same height, Pup (6 units tall) is in the 1-unit hole
#1, Bro (8 units) is in the 3-unit hole #2, Dad (10 units) is in the 5-unit hole #3, Mom (9 units) is
in the 4-unit hole #4, and Sis (7 units) is in the 2-unit hole #5.
Here is another way to solve it:
From the first picture,
Mom = 8 units above ground
Bro = 5
Sis = 2
Pup = 2
Dad = 8
From the second picture,
Mom = 4 units above ground
Bro = 7
Sis = 5
Pup = 3
Dad = 6
Comparing the same people in each hole,
M = B + 1
B = P + 2
S = M - 2
P = D - 4
D = S + 3
Solving simultaneously, you can express everyone in terms of Pup:
P
S = P + 1
B = P + 2
M = P + 3
D = P + 5
Then using some logic, you can determine the depths of the hole and where each must stand.
Correctly solved by:
| 1. James Alarie | Flint, Michigan |
| 2. Kelten Hauf |
Mountain View High School, Mountain View, Wyoming |
| 3. Mike Bova | Wallingford Connecticut |
| 4. Trystin Walker |
Mountain View High School, Mountain View, Wyoming |