Solve Enjoyable Enigmas with Mr.E's Challenge

[TumblingDice]

He could have meant it that way, but I don't think the word "weighting" would have been used correctly if he did. (... Yah-dee-yah-dee-yah... ) The question, if he meant it that way, should have been, "What's the minimum number of times the weights have to be used in dividing the grain into two quantities, one of 4kg and the other of 14kg."

I make allowances for people who put effort into posting/communicating in English, and make my effort to consider the context.

A balance scale has two weights used for weightings: 0.1 kg weight and 0.4 kg weight.

What do you think "weightings" implies here? What are the two weights "used for?"

I have a definition to share, too:

snob (snäb)
noun
One who has an offensive air of superiority in matters of knowledge or taste.

 

[Mark44]

I don't see a spoiler tag in the options now offered above the reply box. Where did you find it?

Also, when I quote your all white posts, there is nothing there unless I highlight it. It's weird.

He's using COLOR=rgb(255, 255, 255), which makes the text pure white. When you highlight the text, it changes to the highlight color.

 

[zoki85]

What do you think "weightings" implies here? What are the two weights "used for?"

By weighting I meant process of determining weight. In this case we use the scale and weights.

 

[TumblingDice]

Exactly, Zoki. Thank you for confirming what I read your post to mean. Enigman tweaked up the answer you were looking for, right?

 

[zoki85]

Exactly, Zoki. Thank you for confirming what I read your post to mean. Enigman tweaked up the answer you were looking for, right?

The following solution is pretty much the same:
Divide the 18 kg between the two pans of the scale. Remove the grain from one pan and divide the other pan's 9 kg of grain between two pans. Now we have 4.5 kg on each pan. Remove 0.5 kg from one of the pans by using both the weights on the other side. You now have 4 kg and 14 kg

 

[zoobyshoe]

The following solution is pretty much the same:
Divide the 18 kg between the two pans of the scale. Remove the grain from one pan and divide the other pan's 9 kg of grain between two pans. Now we have 4.5 kg on each pan. Remove 0.5 kg from one of the pans by using both the weights on the other side. You now have 4 kg and 14 kg

So, how many "weightings" have you performed here?

 

[zoki85]

So, how many "weightings" have you performed here?

3

 

[zoki85]

One more.
Suppose you have four containers: 2, 4, 5, and 9 liters. 9 liters container is full of wine.
Divide the wine into three equal portions in the fewest steps. No spilling!

 

[zoobyshoe]

3

That's what I count as well. So, by "weighting" you meant "weighing," and by that you meant to determine the weight of something using a balance or scale, with or without the use of dedicated, premade weights. Dividing a quantity of something into two equal parts, one side of the balance used to 'weigh' the other, still constitutes a "weighing."

 

[zoobyshoe]

Scrap my previous solution. This is better:

Spoiler

1.) Fill 4 from 9
2.) Fill 2 from 9. This leaves 3 liters in 9.
3.) Pour the 2 liters now in the 2 container into the 5.
4.) Fill 5 to the top from 4. This leaves 1 in the 4 container.
5.) Fill the 2 container from the 5. This leaves 3 in the 5 container.
6.) Pour the two container into the 4 container. The 4 container now has 3.

 

[zoki85]

Scrap my previous solution. This is better:

Spoiler

1.) Fill 4 from 9
2.) Fill 2 from 9. This leaves 3 liters in 9.
3.) Pour the 2 liters now in the 2 container into the 5.
4.) Fill 5 to the top from 4. This leaves 1 in the 4 container.
5.) Fill the 2 container from the 5. This leaves 3 in the 5 container.
6.) Pour the two container into the 4 container. The 4 container now has 3.

Yes, these are the steps ;)

 

[zoobyshoe]

Yes, these are the steps ;)

Thanks!

(I was all the way down at the coffee shop when I realized my first answer had superfluous steps. I rushed back, hoping no one had posted in the meantime.)

 

[zoobyshoe]

I thought this one was hard due to ambiguities that required me to calculate two separate answers. If you don't perceive any ambiguity it might be easy:

Customs Duties

Two wine merchants arrive at the gates of Paris. One has 64 and the other 20 barrels of wine. Since they have not enough money to pay the customs duties, the first pays 40 francs and five barrels of wine, and the second pays 2 barrels of wine but receives 40 francs in change.

What is the value of each barrel of wine, and what is the duty payable on it?

 

[zoki85]

Spoiler

Not sure if there is catch somewhere but
64x=5y+40
20x=2y-40

Value of a barrel = 120 francs
Duty payable on a barrel = 10 francs

 

[zoobyshoe]

Spoiler

Not sure if there is catch somewhere but
64x=5y+40
20x=2y-40

Value of a barrel = 120 francs
Duty payable on a barrel = 10 francs

I think this is a perfectly viable answer. There is, however, an alternate answer.

 

[zoki85]

Spoiler

Dunno, maybe they had 69 barrels and 22 barrels respectively before paying duty?

 

[zoobyshoe]

Spoiler

Dunno, maybe they had 69 barrels and 22 barrels respectively before paying duty?

No, this isn't it. The number of barrels given is exactly what they arrived at the gates of Paris with.

I'll let this sit a while and see if anyone else hits on the alternative answer.

 

[zoki85]

Spoiler

Ok, my last try on it. If the barrels they are paying with are considered money and not goods, they are actually paying duty for 59 barrels and 18 barrels respectively. So the equations would be:
59x=5y+40
18x=2y-40
?

 

[zoobyshoe]

Spoiler

Ok, my last try on it. If the barrels they are paying with are considered money and not goods, they are actually paying duty for 59 barrels and 18 barrels respectively. So the equations would be:
59x=5y+40
18x=2y-40
?

So, looking at this new way, what is the value of each barrel of wine, and what is the duty payable on it?

 

[zoki85]

So, looking at this new way, what is the value of each barrel of wine, and what is the duty payable on it?

That doesn't interest me anymore now if you confirm this way that was the catch;)

 

[zoobyshoe]

That doesn't interest me anymore now if you confirm this way that was the catch;)

There never was a catch. There is an ambiguity. You could assume, as you did, that they were required to pay customs duty on the whole amount they originally wanted to take into Paris, (even though they wouldn't actually be taking that amount in because they were using part of it to pay the duty) or, you could assume they would only be required to pay duty on the amount they actually ended up taking in, 59 and 18 barrels, respectively.

The problem is that either way, the amount they end up paying, and the number of barrels they end up taking into Paris, are exactly the same, despite the fact the price per barrel and the duty amount charged per barrel are different in the two different scenarios. The puzzle, as its stated (and I copied it verbatim from the book) has no indication of which of these two possibilities it is looking for. It makes a difference because they specifically ask for the price per barrel and the amount of duty.

I didn't see any way around it but to calculate both possibilities and then check to see which the book wanted.

 

[zoobyshoe]

Using ten coins, construct a triangle like this:

Now, completely invert the triangle by moving only three of the coins.

 

[zoki85]

Spoiler

 

[zoobyshoe]

Spoiler

Correct! And nice graphic work!

 

[zoobyshoe]

There are 5 sacks, of which Nos. 1 and 2 weigh a total of 12 lbs; Nos. 2 and 3, 13 1/2 lbs; Nos. 3 and 4, 11 1/2 lbs; Nos. 4 and 5, 8 lbs; Nos. 1, 3, and 5, 16 lbs.

Find the weight of each sack.

 

[Enigman]

Spoiler

5.5, 6.5, 7, 4.5, 3.5

 

[zoobyshoe]

Spoiler

5.5, 6.5, 7, 4.5, 3.5

Yes.

Could you describe how you found the answer? The way I did it (which got me the right answer) seems remote from the method described in the book, and I can't even understand the book's procedure.

 

[Enigman]

I did it using Matrix method of solving equations.
http://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html

 

[zoobyshoe]

I did it using Matrix method of solving equations.
http://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html

Thanks. I'll have to study that a while to see if I can make sense out of it.

Here's what the book said:

Spoiler

"The sum of all the weighings, 61 lbs, includes sack 3 thrice and each of the others twice. Deducting twice the sum of the first and fourth weighings, i.e. 21 lbs for thrice 3 - i.e. 7 lbs for sack 3. The rest follows."

This enigma, incidentally, was written by Lewis Carroll

 

[zoki85]

Cute little puzzle for people who like chess.
White has just made the last move and the game ended in a draw (stalemate-see diagram). Determine that move.

 

[collinsmark]

Cute little puzzle for people who like chess.
White has just made the last move and the game ended in a draw (stalemate-see diagram). Determine that move.

Spoiler

King (at d1) takes knight at c1.

[Edit: that was a toughy! I was halfway convinced for awhile that this enigma did not have an answer. But there is one! :w]

 

[mfb]

Spoiler

The black king must have moved all the way below the pawns at some point. As those did not move, A3/B3 were always blocked. It would be tempting to assume the knight moved in, but then the black king would have had to come from A1, which was not a legal position before. Therefore, the black king did not move last time, and the white king captured a black figure at C1. The king was at D1 before, this just allows a black knight to move in. The capture scenario is not possible with the white knight as there is no possible way the black figure could have moved to A1 before.
Summary: Black knight from somewhere moves to C1, white king from D1 captures it.
This was a bad decision, taking the rook would have been better.[/size]

Edit: Wow, took me way too long to write that answer. Well, with proof this is the only option.

 

[zoki85]

That's right, king takes knight on C1 :)

 

[zoobyshoe]

That's right, king takes knight on C1 :)

This puzzle was beyond me, and I'm amazed mfb and collinsmark were able to figure out the last move involved taking a specific piece which is no longer even on the board. Are they PSYCHIC?

 

[drizzle]

They must be.oo)

 

[zoki85]

This puzzle was beyond me, and I'm amazed mfb and collinsmark were able to figure out the last move involved taking a specific piece which is no longer even on the board. Are they PSYCHIC?

They must be.oo)

Haha, doing a little of retrograde analysis certainly helps more here

 

[zoki85]

If you like this sort of chess puzzles, I'll post one more:

White just played a move (diagram), but instantly noticed that wasn't the best move.
With allowance of Black, White takes back the move, and then executes a new move giving a checkmate with it.
Can you reconstruct the moves?

 

[consciousness]

Nice one

Spoiler

White took on black's bishop on e8 with his pawn promoting to a knight. Best was pawn to f8 knight promotion resulting in mate.
[\spoiler]

 

[zoobyshoe]

White just played a move (diagram), but instantly noticed that wasn't the best move.
With allowance of Black, White takes back the move, and then executes a new move giving a checkmate with it.
Can you reconstruct the moves?

I'm confused. Does the image depict the situation just after white made the move it regretted, or does it depict the situation before white made the move it regretted?

 

[zoki85]

Nice one

Spoiler

White took on black's bishop on e8 with his pawn promoting to a knight. Best was pawn to f8 knight promotion resulting in mate.
[\spoiler]

Correct:)

 

[TumblingDice]

White just played a move (diagram), but instantly noticed that wasn't the best move.

I'm confused. Does the image depict the situation just after white made the move it regretted, or does it depict the situation before white made the move it regretted?

I think the image is, "just after white made the move it regretted." :( That's based on zoki85 indicating, "White just played a move (diagram)."
- TD

 

[consciousness]

It is known that a quadratic equation has either 0, 1, or 2 unique real solutions. Well, look at this equation:
$$\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}=1$$
Without loss of generality, assume a < b < c. Now note that x=a, x=b, and x=c are all unique solutions! How can this equation have 3 solutions?!

 

[mfb]

Spoiler

1=1 has more than 2 solutions.

Very nice puzzle.

 

[consciousness]

Correct.

Spoiler

If a quadratic has more than two solutions then it will be an identity and therefore have infinite solutions.