-1. Can a ball with a diameter of 7 cm pass through the gap between the ball, ground, and wall? Please explain your answer.



-2. How many times a day does a digital clock show digits that can be read the same forward and backward? For example, 01:10 Note that the clock displays time on a 24-hour scale.


-3. A young boy was carrying buckets of water from the well. When the bucket was 1/3 full, it weighed 3.5 lb. When it was full, it weighed 6.5 lb. How heavy is the bucket?

3.
When the bucket was 1/3 full, it weighed 3.5 lb. [1]
When it was full, it weighed 6.5 lb. [2]
How heavy is the bucket?

b = bucket weight
w = water weight

b + w/3 = 3.5
3b + w = 10.5 [1]

b + w = 6.5
w = 6.5 - b [2]

[2] in [1]:
3b + 6.5 - b = 10.5
2b = 10.5 - 6.5
2b = 4
b = 2

https://www.youtube.com/watch?v=045__ojHb_g

Extra credit for Denis!

1. Yes it can.

Distance from centre of large ball to corner of floor and wall = sqrt(400+400) = 28.28 cm (good ol' Pythagoras)
Large ball has radius of 20 cm, leaving a gap of 8.28 cm for the small ball to pass through

Good start, and it feels right, but nope.

Now I look at it, can see what you mean . Hm - but how to prove it...

I'll start you off. The 20 cm radius circle has it's center 28.28 cm from the corner, as you showed. How about the 7 cm circle?

Gotcha

The smaller circle's centre would be sqrt(3.5^2 + 3.5^2) = 5 cm away from the corner. This means its outer edge would be 5 + 7/2 = 8.5 cm away, more then the allowable 8.28. So - no-go

So no-go!

2. I'm going to go with 16

0000 0110 0220 0330 0440 0550 1001 1111 1221 1331 1441 1551 2002 2112 2222 2332

And just in time, Cue!

I like #1: easy to jump to conclusions!

Exact answer:
diameter = 40(3-2√2) = ~6.86, so poor "7" is outta luck!