I came up with these questions I thought would be easy to solve, but apparently I can't find how to!

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?

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.

Hats off to anyone who can solve either!

MathProblemSolver101 Jan 22, 2022

#1**+4 **

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....

Guest Jan 22, 2022

#2**+4 **

Solution:

Jack’s fist-half percentage is 75% with 15 successes,

Jack’s second-half percentage is 100% with 10 successes

Jack’s __total percentage__ is 83.333% with 25 successes in 30 shots.

Jill’s __total percentage__ is also 83.333% with 25 successes in 30 shots.

Jill’s fist-half percentage has to be __less than__ 75% for 12 shots.

So, 0.75*12 = 9. Jill had 8 successes in 12 attempts (66.666%).

Then 25 - 8 = 17, so Jill had 17 successes in 18 attempts (94.44%).

Each of these percentages is less than Jack’s percentages for the first and second half.

Jill had (8) successes in the first half and (17) successes in the second half.

GA

--. .-

GingerAle
Jan 22, 2022

#3**+4 **

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

Melody Jan 22, 2022

#6**+2 **

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?

MathProblemSolver101 Jan 23, 2022

#7**+4 **

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?**

Melody Jan 23, 2022

#8**+2 **

I can try

You can have something like

{triangle} {triangle} {triangle}

{square} {square} {square}

{triangle} {triangle} {square}

MathProblemSolver101
Jan 23, 2022

#10**+2 **

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?

=^._.^=

catmg Jan 23, 2022

#12**+3 **

Thanks for clearing the problem up... this turned out to be correct and all the steps are fairly logical. also I realized that diagonal was impossible..because then you cant construct a line of different shpae

MathProblemSolver101
Jan 27, 2022