Math Fun

-1. The economy of Roo Island is rather strange. They have four coins, each worth a different number of Neopoints. Three red coins plus two Neopoints equal a green coin. Three green coins plus five Neopoints equal a purple coin. Eleven red coins plus a single Neopoint equal a purple coin.

The fourth coin is a silver coin that is worth four purple coins. If I go to the Roo Island bakery and buy a loaf of bread worth 84 Neopoints with my silver coin, and the shopkeeper gives me change in red coins... how many red coins would I get?

-2. How can you draw a square with only 3 straight lines?

-3. What 3-digit number is described below?

If my three were a four
And my one were a three
What I am would be nine less
Than half what I'd be

-4. A bug starts out on the inside of a box that is 3 by 4 by 5. He is at height 1 from the base and distance 1 from the nearest corner along the side of length 4 (up along the side facing you). He wants to get to the corner diagonally opposite from his nearest corner at the base of the box. He cannot lose contact with the inside surface. What is the length of his shortest path?

Three red coins plus two Neopoints equal a green coin.
3r + 2 = g : g - 3r = 2 [1]

Three green coins plus five Neopoints equal a purple coin.
3g + 5 = p : -3g + p = 5 [2]

Eleven red coins plus a single Neopoint equal a purple coin.
11r + 1 = p : 11r - p = -1 [3]

The fourth coin is a silver coin that is worth four purple coins.
s = 4p [4]

Solving [1], [2] and [3] (3 unknowns, 3 equations) results in:
r = 5, g = 17 and p = 56

[4]: s = 4 * 56 = 224

If I go to the Roo Island bakery and buy a loaf of bread worth
84 Neopoints with my silver coin, and the shopkeeper gives me
change in red coins... how many red coins would I get?
s - 84 = 224 - 84 = 140
140 / 5 = 28 = number of red coins you'll get.

Well done, Denis.

2. Draw it at the edge of the sheet of paper, so that the edge provides the fourth side of the square (or is that cheating?).

4. If I'm interpreting the bug's location correctly,
I think the shortest path is 3*sqrt(5).

4. I'll have a go at sqrt (7^2 + 5^2) = 8.6

No correct answer yet. If the box wasn't see-through, the bug would be visible on the end flap facing you. Hope this helps.

2: expanding on Lex's sheet of paper:
can be done with 2 straight lines!

On #4:
"He is at height 1 from the base and distance 1 from the nearest
corner along the side of length 4 (up along the side facing you)"

That seems impossible: a diagonal is required from the "x" to the
nearest corner, which has to be greater than 1, since height = 1.
What am I missing?

Anyhoooo...bug travels directly to bottom: that's 1.
Then travels directly to far corner: path is hypotenuse
of right triangle with legs 3 and 5, so sqrt(3^2 + 5^2) = ~5.83
So total = 1 + 5.83 = 6.83 : a bit longer than CC's 3sqrt(5) = ~6.71

Gaaah! CCb was right! I miscalculated 3*sqrt(5) in my head.

It's easiest to lay the end of the box out flat, as shown below.



The bug travels the hypotenuse of a right triangle with legs of 3 and 6.

So the bug being 1 from the corner is not correct;
should be 1 from the side...right?

Ashamed to say that, my spatial vision being terrible,
I massacred a cereal box to build a replica

He is where I marked the little x.

Great use of the cereal box!

2. Draw a square and put only 3 straight lines in it!

#3. 4 possibilities; n = missing digit:

1: 13n -> 34n [1]
2: 1n3 -> 3n4 [2]
3: 31n -> 43n [3]
4: 3n1 -> 4n3 *
5: n13 -> n34 [4]
6: n31 -> n43 *
* these 2 are eliminated (division by 2)

[1] 130 + n + 9 = (340 + n)/2 : n = 62
[2] 100 + 10n + 3 + 9 = (300 + 10n + 4)/2 : n = 8
[3] 310 + n + 9 = (430 + n)/2 : n = -208
[4] 100n + 13 + 9 = (100n + 34)/2) : n = -1/10

So [2] is da winnah: 183
183 + 9 = 192 : 384/2 = 192

This has gotta be the most "fun" algebra problem I've seen!!

Sherlock

Denis, well done!

2. draw a square and then draw 3 lines inside it

Nope, that's what Sherlock suggested. I will admit that this one is worded sneakily.

one of the lines is not quite straight?

Hi Sherlock, great minds think alike

square with 3 straight lines

I was wondering about a sneaky capital A to the power of 2 (therefore squared)

I have a really great hint for you but I'm loving the group think tank too much. I'll post my hint tonight if nobody solves it first. You're so close!

You are cooking a date square and you draw it
out of the oven using a 3-prong fork

Hint time! And Lex already touched on it.

There's more than one kind of square in math.

Denis, very creative, but not correct.

4

Nailed it!

(I hear the sound of palms slapping foreheads )

VERY cute puzzle!

Another solution: 1 (1^2 = 1)

|\/ = 4 and |X = 9 and [ = 100