Prospect Generalist Enigmas And Puzzles

[professorplum13]

Qu 7. Frosty had 4 identical spherical snowballs. He places 3 on the level ice and the 4th on top so that each touched the other 3. The stack height was 16 inches. What was the radius of the snowball?
Can someone please guide me to how to work this out?
Vol of sphere = 4/3 pi r^3 ... etc

Qu 5 In how many ways can 6 of santa's reindeer be arranged in a line so that Dasher is next to Dancer and Donner is next to Blitzen?

[Svejk]

5. There are 57 combinations in a 6 horse bet. Not sure if that is applicable here.

[Svejk]

5. There are 57 combinations in a 6 horse bet. Not sure if that is applicable here.

[fiction-factory]

I calculate the diameter of each ball to be circa 8.808 inches, so the radius of each ball is 4.404 inches. The exact answer though should have root signs in it but I'll need to wait until I have more time

[professorplum13]

Thank you.
fiction-factory - how did you calculate it please? BTW your answer is one of the possibles!

[fiction-factory]

I calculate the exact radius to be 8(3- sqrt 6)

[fiction-factory]

so as a decimal that's 4.404082064

[professorplum13]

Thank you ff - how did you do it?

[fiction-factory]

I did it by using the rule that the total height of the stack in a tetrahedron is d + (n-1)d√(2/3) where n is the number of balls along the longest row of each base and d is the diameter of each ball.

In this case the three balls on the base must be an equilateral triangle so n=2.

I then worked back from the height of 6 to find d.
I then halved it to get r

I'm sure there's an easier way using the volume of a sphere calculation but couldn't visualise it. I'll need to dig out 4 tabletennis balls

[Jonathan-Joe]

Qu 5:
Assume to start that Dasher and Dancer together occupy one position, similarly Donner and Blitzen. We now have 4 positions to fill; this can be done in 4 x 3 x 2 x1 = 24 distinct ways.
Now, for each arrangement Dasher and Dancer could exchange places thus doubling the total number, similarly Donner and Blitzen, doubling again.
So total number of arrangements 24 x 2 x 2 = 96.

[professorplum13]

Thank you ff for that explanation - i can just about follow it!
& Thank you to Jonathan Joe for your very logical explanation.

[fiction-factory]

This may help as it shows how to calculate the height of a stack from a given diameter
http://www.had2know.com/academics/pyramid-of-balls-height-calculator.html

[professorplum13]

Thanks ff - I was just studying that page when you posted!