Two Meticulous Questions

Jack and Jill went up the hill

To fetch a pail of water.

Jack fell down and broke his crown,

And Jill came tumbling after.

When the fun was done

Up Jack got and down he trot

As fast as he could caper;

And went to bed and covered his head

In vinegar and brown paper.

When Jill came in how she did grin

To see Jack's paper plaster;

Mother vexed, did whip her next,

For causing Jack's disaster.

Now Jack did laugh and Jill did cry

But her tears did soon abate;

Then Jill did say that they should play

At see-saw across the gate.

The next day, Jack and Jill when up the hill to shoot paintballs at a basket.

After they finished playing they solved this problem.

Solution below....

1. In a paintball game, Jack attempted 20 shots and made 15 in the first half. In the second half, he was a marksman and made every single one of his 10 shots.

Jill scored a lower percentage of shots than Jack in each of the halves, but had exact same overall percentage as Jack.

If Jill attempted 12 shots and 18 shots in the first and second half respectively, how many baskets did Jill score in each?

Jack attempted 30 shots and made 25.      25/30 

Jill attempted 30 shots She scored the sam percentage in total  but less spercentage in each half

First half

Jack got 3/4

Jill got less than 3/4 of 12  so less than 9.    That would be a max of 8 out of 12

Second half

Jack got 100% so Jill go less than that, a maximum of  17 out of 18

\(\frac{8-c+17-k}{30}=\frac{25}{30}\\ 25-(c+k)=25\\ c+k=0\\\)

So Jill got 8 in the first half and 17 in the second half

Thank you melody and GA for your thoughtful answers! I made some progress on 2, considering combinatorics, but got an answer that I'm sure is incorrect or overcounted for: \(\binom{3}{2} \cdot 3! \cdot 3!\). This answer feels wrong, but I'm not sure why - anyone want to take a stab?

2. Consider a 3x3 square with triangles and squares. In how many ways can the square have three triangles in a line and three squares in another line?

(a) Disregard diagonal lines.

(b) Regard diagonal lines.

Do you have a pic to show us what you are talking about?

I'm gonna try giving problem 2 a shot. 

Part a: without diagonal lines 

Note that vertical lines and horizontal lines are the same, so we only need to count for one and multiply it by 2. 

There are 2 cases.

Case 1: a line of triangles, a line of squares, and a line of both triangles and squares. 

For the line of both triangles are squares, there are 6 possible arrangements. (s, t, t) (t, s, t) (t, t, s) (t, s, s) (s, t, s) (s, s, t)

There are also 6 possible arragements of the 3 lines (3!). 

Therefore, there are 36 possible options. 

Case 2: a line of triangles, a line of squares, and a line containing all triangles or all squares. 

For the line containing all triangles or all squares, there are 2 possible arragements. (s, s, s) (t, t, t)

Finally, for each option there are 3 possible arrangements for the 3 lines. (3C2)

Therefore, there are 6 possible options. 

Adding up both the cases we have 42 options. 

However, we have to account for the fact that we can use both horizontal and diagonal lines, so we have a final answer of 84.

Part b: without diagonal lines

I don't think it is possible to have a grid with a diagonal line of squares or triangles since it would make having another line for the other shapes impossible. 

Therefore, the answer is the same as without diagonal lines, 84. 

My answer seems to be different from yours, do you mind explaining your logic?

=^._.^=