Q: If 6 people take 3 days to dig 8 ditches, working at the same rate how long will it take 4 people to dig 10 ditches?

A: 5.625 days

18 people days to dig 8 ditches = 2.25 people days per ditch

10 ditches = 22.50 people days

22.50 / 4 = 5.625 days.

Q: On an analog clock, how many times do a clock’s hands overlap in a one week period?

A: 154
At 12:00, the hands exactly overlap but not at other times — for example, at 1:05, the hour hand has moved slightly… and at 6:30, the hour hand is halfway between the 6 and the 7.

Making the times that they will overlap only 11 times in 12 hours (60 minutes /11).

12:00, 1:05ish, 2:10ish, 3:15ish, 4:20ish, 5:25ish, 6:30ish, 7:35ish, 8:40ish, 9:45ish, 10:50ish.
In one day they will overlap 22 times so in one week, 154 times.

A wooden cube that is 20 cm on each side is composed of 1 cm x 1 cm x 1 cm  cubes.
Q1: If you paint all 6 sides of the outside of the cube blue, how many cubes will have no blue paint?
With a 3 by 3 by 3 cube, there is just one cube in the middle that would have no paint.
With a 4 by 4 by 4 cube, there are 8 cubes in the center of the cube (2 x 2 x 2) with no paint.
With a 5 by 5 by 5 cube, there are 27 cubes in the center of the cube (3 x 3 x 3) with no paint.

Answer: The pattern continues so a 20 by 20 by 20 cube, there would be 18 x 18 x 18  (5832) cubes in the center of the cube with no paint.

Q2: What if the 20 x 20 x 20 cm cube is composed of 2 cm x 2 cm x 2 cm cubes?
Answer:  There would now be 1000 cubes (10 x 10 x 10), there would be 8 x 8 x 8 (512) cubes in the center of the cue with no paint.