In Kakuro (also known as Cross Sums), no digits can be repeated in each sum. For example, 16 for two boxes must be 9 and 7 (or 7 and 9) and cannot be 8 and 8.

# Category Archives: brain teaser

# October 2019 Brain Puzzler Solution

Q: How many factors of 155^9 are perfect squares and/or perfect cubes?

A: 37 factors of 155^9 are perfect squares and/or perfect cubes.

155 is a composite number composed of 5 x 31.

Therefore

155^9 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 31 x 31 x 31 x 31 x 31 x 31 x 31 x 31 x 31

**25 Perfect Squares**:

1: 1 is both a perfect square and a perfect cube.

4: Perfect squares with just 5s = 5^2, 5^4, 5^6, 5^8

4: Perfect squares with just 31s = 31^2, 31^4, 31^6, 31^8

Combining the 5s and the 31s Perfect Squares

4: 5^2 x 31^ 2, 5^4 x 31^2, 5^6 x 31^2, 5^8 x 31^2 **
**4: 5^2 x 31^ 4, 5^4 x 31^4, 5^6 x 31^4, 5^8 x 31^4

**4: 5^2 x 31^ 6, 5^4 x 31^6, 5^6 x 31^6, 5^8 x 31^6**

**4: 5^2 x 31^8, 5^4 x 31^8, 5^6 x 31^8, 5^8 x 31^8**

**12 Perfect Cubes**:

Perfect cubes with just 5s = 5^3, 5^6 (already counted above), 5^9

**(2)**

Perfect cubes with just 31s = 31^3, 31^6 (already counted above), 31^9

**(2)**

Combining the 5s Perfect Cubes and the 31s Perfect Cubes

5^3 x 31^3, 5^6 x 31^3, 5^9 x 31^3 **(3)
**5^3 x 31^6, 5^9 x 31^6

**(2)**

5^3 x 31^9, 5^6 x 31^9, 5^9 x 31^9

**(3)**

# September 2019 Brain Puzzler Solution

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.

# August 2019 Brain Puzzler Solution

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.

# July 2019 Brain Teaser Solution

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.