Math can be pursued in many ways.  One way is (sometimes disparagingly) called 'recreational mathematics' including games and puzzles.  There are thousands and thousands of math puzzles.  These come at all levels of difficulty, and all levels of education (the two aren't synonymous by any means)

Today, I share a few of my favoirtes, and invite you to share yours.

First, some ground rules:
1.  I will supply answers in tomorrow's diary
2.  If YOU want to supply an answer, or ask about one, go ahead, but PLEASE put it in your title so that people who don't want to see don't do so by accident.
3.  I encourage people to submit their own favorites.  But PLEASE give it a name, and indicate if any advanced math is neeeded, and whether it has a known solution

OK - enough.  On to the puzzles

Puzzle 1 - Two coins - No advanced math.  Known solution.
You have two ordinary US coins in your hand.  The total value is 30 cents.  One of the coins is not a quarter.  What are the two coins?

Puzzle 2 - Counterfeit coins.  No advanced math.  Known solution.  You have 5 bags of 200 coins each.  One of them contains counterfeit coins.  The others contain real ones.  Counterfeit coins each weigh 90 grams.  Real coins each weigh 100 grams.  You have a scale (a regular scale that give a weight, not a balance), assume it is very accurate.  Using the scale only once, how can you tell which bag has the counterfeit coins?

Puzzle 3 - More counterfeit coins.  No advanced math.  Known solution.  Same as puzzle 2, except now you don't know how many bags have counterfeit coins.  How many times do you have to use the scale?

Puzzle 4 - Chicken and egg.  5th or 6th grade math.  Known solution.  If a hen and a half can lay an egg and a half in a day and a half, how many eggs can one hen lay in one day?

Puzzle 5 - Letters. No advanced math.  Known solution.
What comes next in this sequence?
OTTFFSSE

Puzzle 6 - xyz.  At least algebra, maybe more.  Might have a solution, I don't know.

Either solve this, or prove it can't be solved:
x + y + z = 1
x^2 + y^2 + z^2 = 2
x^3 + y^3 + z^3 = 3

Puzzle 7 3n + 1.  Maybe advanced math.  No known solution.

Take any positive integer (counting number).  If it's odd, multiply by 3 and add 1.  If it's even, divide by 2. Repeat until you get to a loop.. For example, if you start with 7 the sequence is

7  -  odd - 3N + 1 = 22
22 - even - half =  11
11 - odd - 3n+1 = 34
34 even half = 17
then
52  26  13  40   20  10  5   16   8   4  2   1

it's been tested for a LOT of numbers (into the billions) but it hasn't been shown to ALWAYS work, and I don't think they know much about how long a number takes to get to 1

Have fun!

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#### Comment Preferences

• ##### Tip jar(11+ / 0-)

I will be here for about 45 minutes, then in and out the rest of the morning.

If you'd like to leave a tip, a solution, a puzzle, or  a suggestion for a column, this is the place. If you'd like to WRITE a column, let me know.

Republicans believe government is the enemy. When they're in charge, they're right

• ##### not a math comment, but(2+ / 0-)
Recommended by:
42, plf515

you do realize that "in and out" has meanings other than you intended, and that keith o took advantage of this when "reflecting" upon Paris Hilton's arrest for DWI - she wanted and In n Out Burger, and keith made a reference to the connection of MS Hilton with the expresion "in and out."

Those who can, do. Those who can do more, TEACH!

[ Parent ]

• ##### That actually hadn't occurred to me(0+ / 0-)

but thank for pointing it out :-)

Republicans believe government is the enemy. When they're in charge, they're right

[ Parent ]

• ##### I would like to write a column(1+ / 0-)
Recommended by:
plf515

but I'm not sure I'll have the time.  How advanced/deep would it have to be?

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

[ Parent ]

• ##### It's really up to you(0+ / 0-)

I have tried to aim these so that people who had no math beyond high school could grasp them.  Some have been deep, some much less so.

What would you like to write about?

Republicans believe government is the enemy. When they're in charge, they're right

[ Parent ]

• ##### Possibly origami(0+ / 0-)

and how it can be used to do geometry problems.  The main problem is that I'd need to generate images somehow.

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

[ Parent ]

Recommended by:
plf515

Puzzle One: A quarter and a nickel.  The one that's not a quarter is a nickel.

Puzzle Two: Are you sure you mean one weighing and 5 bags?  Oh wait, I know.  Take 1 coin from bag A, 2 from B, 3 from C, 4 from D and 5 from E.  If the final weight is short by 10 grams, it's the one coin from A, it's short by 20 grams, it's the two coins from B, etc.

Puzzle Three: 1 from A, 2 from B, 4 from C, 8 from D and 16 from E.  Say the weighing is short 10 grams.  That must be the one coin from A.  If it's 30 grams short, that must be the 2 from B and the 1 from A.  150 grams would have to be the 1 from A, 2 from B, 4 from C and 8 from D.  (I was actually going to use binary numbers for Puzzle Two just for elegance but decided not to get fancy.  I'm glad I thought of it, since I don't think I've seen this puzzle before.)

Puzzle Four: If a hen and a half lay an egg and a half in any unit of time, one hen can lay one egg in the same unit of time.  The unit of time here is "a day and a half", so a hen can lay an egg in 36 hours.  That means they are 24/36 of the way through an egg after one day, which is 2/3s of an egg.

Puzzle Five: N

Puzzle Six: If x, y and z are all positive there's an easy solution.  Otherwise I'm not sure either.

Puzzle Seven: Hey, I have work to do today, I can't be trying to prove unknown conjectures.

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

• ##### Puzzle 6 - not an answer(0+ / 0-)

There's certainly no answer with all positive integers.
I don't THINK there's a solution with all real numbers.

Republicans believe government is the enemy. When they're in charge, they're right

[ Parent ]

• ##### No answer with all positive reals, either(1+ / 0-)
Recommended by:
plf515

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

[ Parent ]

• ##### these puzzles are pretty old(1+ / 0-)
Recommended by:
plf515

I've seen the first five puzzles many times over the years.  and solutions are available on the net.

But I do like the idea of having puzzles on the site, since you may be able to come up with some I have not seen, or that most people here have not seen.

I only just heard #3 for the first time earlier this year, on Car Talk of all places.  Actually there are a lot of great puzzles on the Car Talk site.

Puzzles without answers aren't really fair on this blog though.  Sir Andrew Wiles didn't have much time for politics.

Politics is like driving. To go backward, put it in R. To go forward, put it in D: TELL THE TRUTH. HOLD THEM ACCOUNTABLE. REPAIR THE DAMAGE. VOTE DEMOCRATIC!

[ Parent ]

• ##### it's an oldie but goodie(0+ / 0-)

I've seen (variations of) puzzle #3 over the years and thought it intriguing but never bothered to work it out.  I never knew the answer until I heard the version on Car Talk earlier this year.

I know the link says the puzzler was aired on May 10, 2004, so maybe I heard a repeat show this past winter.

Politics is like driving. To go backward, put it in R. To go forward, put it in D: TELL THE TRUTH. HOLD THEM ACCOUNTABLE. REPAIR THE DAMAGE. VOTE DEMOCRATIC!

[ Parent ]

• ##### Puzzle 1 and 2(1+ / 0-)
Recommended by:
plf515
1. A quarter and a nickel.  "One of the coins" isn't a quarter ... but the other one is.
1. Get 15 coins: 1 from the first bag, 2 from the second, and so forth.  If the weight is 1490, bag 1 is counterfeit.  If 1480, bag 2.  1470, bag 3.  1460, bag 4.  1450, bag 5.
• ##### A recent favorite(2+ / 0-)
Recommended by:
Brian B, plf515

A circular pizza comes in a square box and the edges of the pizza fit right against the sides of the box.  A business card is 70 mm high by 140 mm long and fits exactly into the corner of the box with one corner just resting against the crust of the pizza.  How big is the pizza?

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

• ##### Puzzle 5: N(1+ / 0-)
Recommended by:
plf515

Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

• ##### Sudoku Baby!!!!(1+ / 0-)
Recommended by:
plf515

The Washington Post has a great interactive Sudoku puzzle.

Well, ok, it's a syndicated puzzle - but Sudoku lovers should be aware... I really like the way you can hold down the shift key and put your own clues in the grid squares...

Pen Rai!

• ##### You should try websudoku.com(1+ / 0-)
Recommended by:
Sanuk

It has some pretty nice optional settings...

I don't know why we think, just because we're mighty, that we have the right to try to substitute might for right -- Wayne Morse (D-Ore), 1964

[ Parent ]

• ##### You guys are good!(1+ / 0-)
Recommended by:
TrueBlueMajority

Puzzle 3 is my own invention, although someone else may have also invented it.

Republicans believe government is the enemy. When they're in charge, they're right

• ##### oops I meant to put(0+ / 0-)

Politics is like driving. To go backward, put it in R. To go forward, put it in D: TELL THE TRUTH. HOLD THEM ACCOUNTABLE. REPAIR THE DAMAGE. VOTE DEMOCRATIC!

[ Parent ]

• ##### Puzzle 3(1+ / 0-)
Recommended by:
plf515

Again, the answer should be one trial.

Bag 1: 1 coin [-10 if fake]
Bag 2: 2 coins [-20]
Bag 3: 4 coins [-40]
Bag 4: 8 coins [-80]
Bag 5: 16 coins [-160]

[These are not the only numbers that'll work, but they're the easiest.]

• ##### The unsolved one is interesting(2+ / 0-)
Recommended by:

Lacking the mathematics, simulate, is my motto. I wrote a short program to test the integers from 1 to whatever, recording the number of iterations required to reach 1. I figured it would look kinda random but was surprised that it seems to have some structure:

In any given range the results jumps between a smallish number of values, rather than any old value between 1 and 300. Odd. For certain stretches, the value is the same for many starting values in a row (the unbroken horizontal stretches above). For example, these starting numbers, 514-517, all take 123 iterations to get to 1, but by different routes for the first 18 iterations:

1. 257 772 386 193 580 290 145 436 218 109 328 164 82 41 124 62 31 94...
1. 1546 773 2320 1160 580 290 145 436 218 109 328 164 82 41 124 62 31 94...
1. 258 129 388 194 97 292 146 73 220 110 55 166 83 250 125 376 188 94...
1. 1552 776 388 194 97 292 146 73 220 110 55 166 83 250 125 376 188 94...

Thing blow up at 113,383. Even after 100 million iterations, it didn't reach 1.

• ##### Way cool!(0+ / 0-)

At least to a geek like me

Republicans believe government is the enemy. When they're in charge, they're right

[ Parent ]

• ##### that graph isn't right(1+ / 0-)
Recommended by:
plf515

5000, for instance, takes more than one different number of iterations to reach 1.

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

[ Parent ]

• ##### Not really(1+ / 0-)
Recommended by:
plf515

It's just that the plotted points are so big that they seem to overlap even when there are gaps between them.

• ##### Here's a blowup around 5000(2+ / 0-)
Recommended by:

It looks like the value jumps around between about a dozen values in this stretch of 100 numbers.

There's gotta be a Fibonacci sequence in there somewhere. ;)

• ##### The curves in the big one look like logs(1+ / 0-)
Recommended by:
plf515

nt

What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

[ Parent ]

• ##### Answers for 1-5(1+ / 0-)
Recommended by:
plf515

These are the easy ones, so here are my answers.

Puzzle 1: Not really a puzzle as much as a joke.  One coin isn't a quarter but the other one is, which makes the first coin a nickel.

Puzzle 2: Take one coin from bag 1, two coins from bag 2, etc., and place them on the scale.  If all the coins were real, the weight would be 1500g.  Since some are counterfeit, the measured weight will be somewhat less than 1500g.  Subtract the measured weight from 1500, then divide by 90 (the weight of one counterfeit coin).  The result will be the number of the bag with the counterfeit coins.

Puzzle 3: Similar to puzzle 2, except this time instead of taking 1, 2, 3... coins take one coin from bag 1, two coins from bag 2, four coins from bag 3, eight coins from bag 4 and sixteen coins from bag 5.  The combined weight would be 3100g if all coins were real.  Subtract the measured weight from 3100 and divide by 90.  Convert the result into a five-digit binary number - all the digits that are 1 represent counterfeit bags (for example: the number 01101 would mean bags 1, 3 and 4 are counterfeit).

Puzzle 4: 2/3 of an egg.  If 1-1/2 hens lay 1-1/2 eggs in 1-1/2 days, one hen lays one egg in 1-1/2 days, which means the hen will have lain 2/3 of an egg in one day (1/3 of an egg every 1/2 day).

Puzzle 5: NTETTFFSSENT.

I'm still working on puzzles 6 and 7...

I don't know why we think, just because we're mighty, that we have the right to try to substitute might for right -- Wayne Morse (D-Ore), 1964

• ##### Oops(1+ / 0-)
Recommended by:
plf515

I made a mistake in 3 and 4.  It should be divide by 10 which is the differenc between a good coin and a counterfeit coin.

Oh well - that's what I get for not working the math out on paper...

I don't know why we think, just because we're mighty, that we have the right to try to substitute might for right -- Wayne Morse (D-Ore), 1964

[ Parent ]

• ##### Not an answer for 6(1+ / 0-)
Recommended by:
plf515

But I find this rather interesting

If x +y +z = 1, then x = 1 -y -z.  Substituting in the other two equations produces the following:

(1 -y -z)^2 +y^2 +z^2 = 2
1 -y -z -y +y^2 +yz -z +yz +z^2 +y^2 +z^2 = 2
2(y^2 +yz +z^2 -y -z) = 1
y^2 +yz +z^2 -y -z = 1/2

and

(1 -y -z)^3 +y^3 +z^3 = 3
1 -y -z -y +y^2 +yz -z +yz +z^2 -y +y^2 +yz
+y^2 -y^3 -y^2z +yz -y^2z -yz^2 -z +yz +z^2
+yz -y^2z -yz^2 +z^2 -yz^2 -z^3 +y^3 +z^3 = 3
3(y^2 +2yz +z^2 -y -z -y^2z -yz^2) = 2
y^2 +2yz +z^2 -y -z -y^2z -yz^2 = 2/3

Subtracting one from the other gives

y^2 +2yz +z^2 -y -z -y^2z -yz^2 = 4/6
-y^2 -yz -z^2 +y +z                    = -3/6

yz -y^2z -yz^2 = 1/6
(1 -y -z)yz = 1/6
xyz = 1/6

I don't know why we think, just because we're mighty, that we have the right to try to substitute might for right -- Wayne Morse (D-Ore), 1964

• ##### Problem 6(1+ / 0-)
Recommended by:
plf515

The second equation puts an upper bound on x, y and z.

x^2 + y^2 + z^2 = 2

Since the square of any number is positive, all the terms in the sum are postive, so none of them can be over 2. Therefore the absolute values of x, y and z must each be less than the square root of 2, i.e. 1.41421.

That allows a numerical check for a solution, trying all values of x and y between -1.41421 and +1.41421 with some suitably small step, e.g., 0.001 (given x and y the value for z would be computed from equation 1).

I did that & there appears to be no solution. There are values of x, y and z that satisfy the first two equations, or the first and third, but not all three.

Not as satisfying as a closed-form proof, I realize.

• ##### That would exclude imaginary numbers(1+ / 0-)
Recommended by:
plf515

For real numbers, the square would be positive.  But for imaginary numbers the square would not necessarily be positive.  (The imaginary number 'i' is said to equal the square root of -1, i.e. i^2 = -1.)

I don't know why we think, just because we're mighty, that we have the right to try to substitute might for right -- Wayne Morse (D-Ore), 1964

[ Parent ]

• ##### That's cool(0+ / 0-)

approach.

But, of course, only applies to the real numbers....

Republicans believe government is the enemy. When they're in charge, they're right

[ Parent ]

• ##### I figured complex numbers were excluded(1+ / 0-)
Recommended by:
plf515

Because the concept of odd and even doesn't exist for complex numbers. So the odd/even rule can be applied.

Is 3 + 3i odd or even?