Solve Enjoyable Enigmas with Mr.E's Challenge

[Enigman]

There isn't. You got it.

 

[collinsmark]

Here's one that I've used before, but not in this enigma context.

Find the Flaw!

There is a flaw in the following mathematics. Find the flaw, state which line contains the flaw, and specify what the flaw is.

We start with assigning the variable x to a, where a can be constant, or another variable, whatever.

$$1. \ \ \ \ \ \ x = a$$
Add x to both sides of the equation.

$$2. \ \ \ \ \ \ x + x = x + a$$
Simplify the left hand side of the equation.

$$3. \ \ \ \ \ \ 2x = x + a$$
Add "-2a" to both sides of the equation.

$$4. \ \ \ \ \ \ 2x - 2a = x + a - 2a$$
Simplify the right hand side of the equation (a - 2a → -a).

$$5. \ \ \ \ \ \ 2x - 2a = x - a$$
Factor a '2' from the left hand side of the equation.

$$6. \ \ \ \ \ \ 2(x - a) = x - a$$
Divide both sides of the equation by x - a.

$$7. \ \ \ \ \ \ \frac{2(x - a)}{(x - a)} = \frac{x - a}{(x-a)}$$
For each side of the equation, cancel out common terms that are in both the numerator and denominator.

$$8. \ \ \ \ \ \ 2 = 1$$
Obviously, two is not equal to one. Something dreadful must have happened. Can you spot the flaw and where it is?

Hint: I may accept more than one possible answer, as long as that answer is explained well.

 

[consciousness]

Spoiler

When we assume that x=a, then it follows that (x-a)=0. Hence you can't divide by it.

 

[Enigman]

Collinsmark what have you done?

 

[collinsmark]

Spoiler

When we assume that x=a, then it follows that (x-a)=0. Hence you can't divide by it.

Collinsmark what have you done?

Spoiler

Yes, that's the jist of it. It mostly comes down to step 7. I would have accepted anything like,

• "You're dividing by 0 in step 7 (of course that's obvious now after fumbling around and I subtracted x from a). What the hell are you doing!? You dork!? I hate you!" or alternatively,
• "In step 7, whenever we divide something by another something, we must stipulate that the other something is not zero. So here we must stipulate that the other something, x - a is not zero, meaning x ≠ a. But if we were to introduce the restriction in step 7 of x ≠ a, then that introduces a contradiction with step 1. Therefore step 7 is not possible. yada yada. ... " Well something of that sort anyway.

I work with mathematically inclined people on a daily basis (engineers, myself being one of them). We've all struggled with this one the first time we saw it. But even with love for math, which we might share, maybe, sometimes rigor is thrown to the wind. Enigmas like this make me appreciate the mathematicians' rigor. We in engineering don't have to deal with this stuff that often. But I love the fact that it happens.

 

[Enigman]

Given: A standard narrow-neck coke bottle and a straw
Your mission should you choose to accept it:
1) Drink the coke...yes I know hell's freezing over but no one said its going to be easy...(or you could just throw it away...)
2) Use the straw and without touching the bottle lift the bottle up. Wrapping the straw around the bottle is not allowed.
P.S. @collinsmark- micromass has an interesting blog entry about seeming paradoxes of calculus and how mathematical rigour resolves them, you may want to check them out.

 

[Enigman]

https://www.physicsforums.com/blog.php?b=4566
Found it.

 

[Enigman]

Given: A standard narrow-neck coke bottle and a straw
Your mission should you choose to accept it:
1) Drink the coke...yes I know hell's freezing over but no one said its going to be easy...(or you could just throw it away...)
2) Use the straw and without touching the bottle lift the bottle up. Wrapping the straw around the bottle is not allowed.
P.S. @collinsmark- micromass has an interesting blog entry about seeming paradoxes of calculus and how mathematical rigour resolves them, you may want to check them out.

...

 

[Jonathan Scott]

...

As often happens with these puzzles, the question seems a little unclear.

(a) Do you mean that one must not touch the bottle even with the straw?

(b) How much do you mean by "lift"? Lift it clear of a surface? Or raise it a little?

Spoiler

Or raise it within e.g. a bucket of water?

 

[Enigman]

As often happens with these puzzles, the question seems a little unclear.

(a) Do you mean that one must not touch the bottle even with the straw?

Use the straw as you want, just don't wrap it around the bottle.

(b) How much do you mean by "lift"? Lift it clear of a surface? Or raise it a little?

Mid-air...try over a soft surface if you are looking for a more hands-on approach.

 

[lisab]

Spoiler

Bend the straw so it's like an L, stick it in the bottle and try to wedge it in the neck, and lift.

 

[Enigman]

Spoiler

Bend the straw so it's like an L, stick it in the bottle and try to wedge it in the neck, and lift.

You got it!

 

[Enigman]

Now stick the straw into a potato: It generally bends when you poke it...so what do you do?
(I seem to remember something about random opinions and Lisab's potatoes...)

 

[zoobyshoe]

This one needs a diagram (attached):

"The magic numbers for the wheel of fortune are: 34, 42, 43, 50, 51, 52, 59, 60, and 68. Place each of these numbers in the proper circle so that:

1. The three numbers on each straight line equal 153.

2. The numbers in circles ABC, CDE, EFG, and GHA also equal 153.

We have placed the number 59 in a circle to get you started."

No computers allowed!

 

[collinsmark]

This one needs a diagram (attached):

"The magic numbers for the wheel of fortune are: 34, 42, 43, 50, 51, 52, 59, 60, and 68. Place each of these numbers in the proper circle so that:

1. The three numbers on each straight line equal 153.

2. The numbers in circles ABC, CDE, EFG, and GHA also equal 153.

We have placed the number 59 in a circle to get you started."

No computers allowed!

Spoiler

Solved it!

But I won't give it away just yet. It's a pretty rewarding puzzle and I don't want to completely spoil it so soon.

But here are some hints in case you are stuck. (*partial spoiler warning*)

(i) There are two solutions to this puzzle. Each solution is a mirror image of the other along the G-<center>-C axis.
(ii) Process of elimination is involved, but choose smartly. It's not too hard to group each number into one of three categories: (a) on the top, bottom, or right, (b) along a diagonal [but not at center], or (c) center.
(iii) Once you find the only number that can be in the center, you get the number to the right as a freebie, extra special bonus.
(iv) When you find a number that can be either at the top or bottom (but not right), just pick one. Remember from (i) that there are two symmetrical solutions. So just pick one or the other, top or bottom; it doesn't matter.

 

[Enigman]

[STRIKE]Work in progress:[/STRIKE] DONE!
34, 42, 43, 50, 51, 52, 59, 60, and 68

Spoiler

Average value: 51
Diameters:
59, 51, 43
60, 51, 42
68, 51, 34
50, 51, 52
...60
...34...50
59...51...43
...52...68...
...42....

Sectors:
with 59: 60,59,34 ; 42,59,52
with 60: 60,59,34 ; 43,60,50
59 and 60 both belong to two sectors

 

[Enigman]

Now stick a straw into a potato: It generally bends when you poke it...so what do you do?
(I seem to remember something about random opinions and Lisab's potatoes...)

...

 

[zoobyshoe]

You both got it. I don't know about you, but the only way I could see to approach it was to do a lot of tedious subtraction from 153. Eventually I figured out 51 was the only option for the center, then the rest was easier.

 

[Enigman]

You both got it. I don't know about you, but the only way I could see to approach it was to do a lot of tedious subtraction from 153. Eventually I figured out 51 was the only option for the center, then the rest was easier.

Well, the center no. had to be there in all the summations, so it would'nt be too big or too small and averaging out all the numbers would give us a rough idea about where the number is.
The average comes out to be 51...which was lucky, I expected decimals...
So 153-51=102 I just went on subtracting the numbers from 102 to see if the remainder was in the list. Luckily it was, otherwise I would have to check with 50 and 52 depending on the remainder <rinse and repeat>

 

[consciousness]

Now stick the straw into a potato: It generally bends when you poke it...so what do you do?
(I seem to remember something about random opinions and Lisab's potatoes...)

Some possible answers-

Spoiler

Cut/bend the straw so that a small segment of it is in your hand. Now push hard (it won't bend as the segment is too small). Alternatively boil the potato to make it soft.

 

[Enigman]

Some possible answers-

Spoiler

Cut/bend the straw so that a small segment of it is in your hand. Now push hard (it won't bend as the segment is too small). Alternatively boil the potato to make it soft.

No damage either to potato or straw would be nice...
(Menu this evening: Coke and chips)

 

[Enigman]

Now stick the straw into a potato: It generally bends when you poke it...so what do you do?
(I seem to remember something about random opinions and Lisab's potatoes...)

Hint:

Spoiler

It won't work with a straw that has a hole in it

 

[AlephZero]

I don't know about you, but the only way I could see to approach it was to do a lot of tedious subtraction from 153.

Not so much "tedious subtraction" as "counting on your fingers" once you get a start.

Look at the differences between the numbers

34 42 43 50 51 52 59 60 68
  8  1  7  1  1  7  1  8

Symmetrical, no? You need four sets of three containing the same number (to go in the center), and they are
34 51 68
42 51 60
43 51 59
50 51 52

You also need three sets containing 59, and the options are
59 51 43 (you got that one already)
59 52 42
59 60 34
You need two of the four numbers 52 42 60 34 to make a line across the center with 51, and the only possibility (from the first list) is 42 and 60.

Now, fill in the diagram

 

[zoobyshoe]

Not so much "tedious subtraction" as "counting on your fingers" once you get a start.

Look at the differences between the numbers

34 42 43 50 51 52 59 60 68
  8  1  7  1  1  7  1  8

Symmetrical, no? You need four sets of three containing the same number (to go in the center), and they are
34 51 68
42 51 60
43 51 59
50 51 52

You also need three sets containing 59, and the options are
59 51 43 (you got that one already)
59 52 42
59 60 34
You need two of the four numbers 52 42 60 34 to make a line across the center with 51, and the only possibility (from the first list) is 42 and 60.

Now, fill in the diagram

Despite having solved the puzzle, I had a strong suspicion my method was the crudest, and that there had to be a more intelligent approach. I did not naturally notice the symmetry around 51 you point out here, but that would have been the needed insight to a very fast solution.

Now that I see the symmetry, I suspect it is necessary for the puzzle to work as it does. If we were to design a similar puzzle summing to a different number, the choices would have also to be symmetrically arranged around the center number.

 

[Enigman]

Now stick the straw into a potato: It generally bends when you poke it...so what do you do?

Spoiler

Think eggs and bottles...https://www.physicsforums.com/showpost.php?p=4516938&postcount=305

 

[consciousness]

Hint:

Spoiler

It won't work with a straw that has a hole in it

Hmm this might work-

Spoiler

Light a matchstick and station the straw directly above it. This should create a a low pressure region inside the straw and it might collapse into a strip, making the job easy.

 

[Enigman]

Hmm this might work-

Spoiler

Light a matchstick and station the straw directly above it. This should create a a low pressure region inside the straw and it might collapse into a strip, making the job easy.

No the heat's going to make it more squishy...we need it to stop bending...keep thinking air pressure.

 

[zoobyshoe]

No the heat's going to make it more squishy...we need it to stop bending...keep thinking air pressure.

Spoiler

Are you suggesting blowing into one end of the straw while you try and push the other end into the spud?

 

[collinsmark]

Now stick the straw into a potato: It generally bends when you poke it...so what do you do?

Hmm.

Spoiler

This might work, maybe. It reminds me of a trick one can use to make a straw pop pretty loud when snapped with a finger (that process destroys the straw). Anyway, here's a modification of that, that might instead work for this one.

But I have not tried this due to temporary lack of straws and potatoes. It's just a guess.

o Take the straw and bend it in the middle.
o Twist each end around so there is a noticeable, sharp, bend/twist thing around in the middle. Be careful though, you don't want to twist so much that it compromises the straw's integrity.
o Keeping the twist in place, hold one end of the straw between a pair of fingers or between a finger and thumb (the finger and thumb is probably easier).
o Do the same with the the other end of the straw and the other hand.
o Pinch both ends of the straw at all times.
o Move both hands in a circular motion with a axis parallel to left and right. The goal is to place twists on both ends of the straw to trap the air inside. If done right, you should be able to feel the straw bulge a little bit in the middle parts, due to the trapped air.
o Don't twist so much that the straw bursts though. Just inflate it enough so it resists bending.
o Bring your hands together, causing the inflated parts of the straw to bend in half. Now there are two inflated parts of the straw with a sharp bend in the middle.
o Use the sharp bend in the middle to impale the potato. This might take a bit of a balancing act to keep the potato from rolling out from under the straw. It also might require chipping/drilling away at the potato skin at first, perhaps with the sharper "bend" in the straw, just to get a divot going that can then be used to ease the process potato goring.
o Once the straw has sufficiently skewered the potato, perhaps it will hold tight enough to lift the potato?

 

[zoobyshoe]

"Each horizontal row in the grid has the same mathematical relationship. If you can identify the pattern, you will be able to supply the missing numbers in the bottom row."

 

[AlephZero]

Too easy, for us oldies who learned to do arithmetic before there were calculators.

Spoiler

24, in base 7. But the answer is in base 10 of course.

 

[zoobyshoe]

I fit that description, and it took a long time before I saw it. You're just very number-minded.

 

[zoobyshoe]

Even easier, but I found it fun:

Magic Nines

"Each digit from 1 to 9 is used only once. The first and second row of digits are added to obtain the sum in the bottom row.

Rearrange these digits in seven similar boxes so that the sum increases each time by nine. More than one solution is possible."

(I suggest putting lower digits on top.)

 

[Enigman]

Spoiler

Are you suggesting blowing into one end of the straw while you try and push the other end into the spud?

That would work...

Hmm.

Spoiler

This might work, maybe. It reminds me of a trick one can use to make a straw pop pretty loud when snapped with a finger (that process destroys the straw). Anyway, here's a modification of that, that might instead work for this one.

But I have not tried this due to temporary lack of straws and potatoes. It's just a guess.

o Take the straw and bend it in the middle.
o Twist each end around so there is a noticeable, sharp, bend/twist thing around in the middle. Be careful though, you don't want to twist so much that it compromises the straw's integrity.
o Keeping the twist in place, hold one end of the straw between a pair of fingers or between a finger and thumb (the finger and thumb is probably easier).
o Do the same with the the other end of the straw and the other hand.
o Pinch both ends of the straw at all times.
o Move both hands in a circular motion with a axis parallel to left and right. The goal is to place twists on both ends of the straw to trap the air inside. If done right, you should be able to feel the straw bulge a little bit in the middle parts, due to the trapped air.
o Don't twist so much that the straw bursts though. Just inflate it enough so it resists bending.
o Bring your hands together, causing the inflated parts of the straw to bend in half. Now there are two inflated parts of the straw with a sharp bend in the middle.
o Use the sharp bend in the middle to impale the potato. This might take a bit of a balancing act to keep the potato from rolling out from under the straw. It also might require chipping/drilling away at the potato skin at first, perhaps with the sharper "bend" in the straw, just to get a divot going that can then be used to ease the process potato goring.
o Once the straw has sufficiently skewered the potato, perhaps it will hold tight enough to lift the potato?

may work but wholly unnecessary...

https://www.youtube.com/watch?v=8Ewpwylbxec

 

[consciousness]

Monks With Eye Problems PART 2-

There is an island of monks where everyone has either brown eyes or red eyes. Monks who have red eyes are cursed, and are supposed to commit suicide at midnight. However, no one ever talks about what color eyes they have, because the monks have a vow of silence. Also, there are no reflective surfaces on the whole island. Thus, no one knows their own eye color; they can only see the eye colors of other people, and not say anything about them. Life goes on, with brown-eyed monks and red-eyed monks living happily together in peace, and no one ever committing suicide.

These monks will believe anything others say, as long as it isn't certain that they are lying.

Then one day a greedy and evil logician comes to the island. He recognizes that this system is very unstable and decides to take advantage of it. He wants the entire island to himself, but he knows that if he speaks an obvious lie the monks will kill him.

Think of a statement he can make to get all of the monks on the island to kill themselves as quickly as possible.
-------------------------------
This is much harder than the first because there we had to find the consequences of a statement while here we have to make the statement. I suggest trying the first one before this(its on page 35).