Saturday Puzzle

Just thought I'd create the graphic for jd2718's question. The source of the puzzle is unknown to me but I'm sure that I've seen it before. The problem is simple ... given a square of side 1 and four circles each of radius 1 centered at the four corners of the square, what is the area of the "rounded square" in the middle?

For what it's worth, I got $1-\sqrt{3}-\frac{\pi}{3}$, but I can't follow all my scribbling to check it. Too much thinking for a Saturday afternoon.

9 & 33 ... Thursday Tidbits

Problem 9

Difficulty: Not difficult at all.

Problem 33

Difficulty: Just algebra. If all kids took and passed algebra II, this problem would be easy for everyone..

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 09 and answer 33.

Wednesday: 8 and 34

Problem 8

Difficulty: Functions aren't my favorite, but this one held my interest.

Problem 34

Difficulty: Middling. Fascinating application of algebraic theory.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 08 and answer 34.

Two for Tuesday, 7 and 35

Problem 7

Difficulty: simple, if you are clear on the algebra.

Problem 35

Difficulty: Middling.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 07 and answer 35.

Monday, Monday. 6, 36

Problem 6

Difficulty: Fairly easy?

Problem 36

Difficulty: Tough to determine the pattern quickly, but as usual, all kinds of stuff cancels.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 06 and answer 36.

Sunday pair: 5 and 37

Problem 5

Difficulty: ACB is 50°. If that isn't showing, refresh the page. Sorry about the mistake when first posted. Arc lengths, anyone?

Problem 37

Difficulty: Fairly difficult.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 05 and answer 37.

Saturday Pair: 4 and 38

This group is from 2002.

Problem 4

 

Difficulty: Easy.  Of course, the only thing worse than fractions for the average kid is compound percentages. That little difference.

Problem 38

Difficulty: Fairly difficult for your students.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 04 and answer 38.

Two for Thursday, 2 and 40

This group is from 2002.

Problem 2

Difficulty: Easy and fun.

Problem 40

Difficulty: Fairly difficult for your students.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 02 and answer 40.

The daily Two, from 2002: 1 and 41

Starting a new set!  This group is from 2002.  As with the previous ones, I'll post two at a time so you won't be tempted to work through all 41 at once.  You have to eat, you know.  Just looking out for your basic health.

Problem 1

Difficulty: Easy and fun. I really don't get why students freak out over these simple fraction problems. Okay, I do understand ... not enough practice.  So here's another practice problem.

Problem 41

Difficulty: Until you see it, "What?" When you see it, "D'oh!"
Simple geometry and two nifty trigonometric doodads.


Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 01 and answer 41.

Tuesday Child is full of grace ... 20, 21, and 22.

Problem 20

Difficulty: Easy, but fun. A new way to look at averages and the effect of an addition on the average, depending on n.

Problem 21

Difficulty: Linear programming.Any bets this will somehow work out to an integer?

Problem 22

Difficulty: Easy. Once you've got the details right.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 20, answer 21 and answer 21.

Monday's Child has Two: 19 and 23

Problem 19

Difficulty: easy. Notice we didn't specify which color.

Problem 23

Difficulty: Very clever geometry question.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 19 and answer 23.

Delicious pair o'Problems: 18 and 24

Problem 2*3²

Difficulty: easy-ish. A little trigonometric - algebraic manipulation and "Surprise!"

Problem 2³*3

Difficulty: Easy. At least once in every test, in some form or another, the creators work the year into a problem.  This test was for 2007.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 18 and answer 24.

Gruesome Twosome: 16 and 26

Sweet Sixteen

Difficulty: easy-ish. Another SAT-type reasoning question. Fun to watch the students work out the ramifications.

Problem 26

Difficulty: Medium. Is it wrong of me to get really annoyed at this question?  It's not terribly difficult, but it is so ... artificial.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 16 and answer 26.

Daily Problematic Pair: 15 and 27

Day Fifteen ... We're almost there.

Difficulty: pretty easy. Kids will get wrapped up in the P(A) and if-then, but it's an example of a problem that is much easier if you imagine the grid of all possibilities. Definitely a candidate for the SAT.

Problem 27

Difficulty: Medium. It's funny how easily the kids get thrown by an infinite series.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 15 and answer 27.

Daily Think. Problems 13 and 29

Day Thirteen

Difficulty: easy-ish. Mostly, it's an interpretation question.

I love this next problem. So very cool.
Problem 29

i.e., find the shaded area.
Difficulty: Medium.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 13 and answer 29.

Daily Two 11 and 31

First, an easy one.

And then something more complicated, though personally, I'm not a fan of probability questions.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 11 and answer 31.

Daily two puzzles for contemplation. 10 and 32

Day Ten

Difficulty: easy-ish.

Problem 32:

Difficulty: Tricky. It seems easy, but then tries to kneecap you.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 10 and answer 32.

Daily Two. 9 and 33

Day Nine.

Difficulty: easy-ish.

Number 33: 

Difficulty: Hard.
(or as the kids said, OMFG)
It's actually more complicated than hard.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 9 and answer 33.

Daily Two. 8 and 34

Day Eight will make you think a second ... or ten.

Difficulty: easy.

Difficulty: Hard-ish
(if they don't know the 3D Pythagorean theorem and have to work it out.)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 8 and answer 34.

Daily Two. 7 and 35

Problem 7

Difficulty: so easy the kids kept second-guessing themselves. ;-)

Problem 35

Difficulty: Hard (last page)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

answer 7 and answer 35.