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Can you move just one digit to a new location, making this equation true?
Do not move signs, only move one numeral.
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82 - 43 = 1
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I don't know how to type it, but the 4 becomes an exponent of the 3.
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I don't know how to type it, but the 4 becomes an exponent of the 3.
82 - 3^4 = 1
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82 - 3^4 = 1
Yes, that.
I guess the caret wasn't important enough to make it to my phone keyboard. : )
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I don't understand. Can somebody please explain?
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I don't know how to type it, but the 4 becomes an exponent of the 3.
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Hooraay!
82 - 34 = 1
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I don't understand. Can somebody please explain?
In other words,
82 - 34 = 82 - (3)(3)(3)(3) = 82 - 81 = 1
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Alternatively, 82 - 34 = 82 - 3*3*3*3
82 - 81 = 1
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There are 14 wineglasses on a table, seven of which are full of wine, but the other seven glasses are only half full.
Without changing the amount of wine in any glass, can you arrange them into 3 groups, such that each group contains the same total amount of wine?
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82 - 34 = 1
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Now I get it, thanks Neil. Hooraay!
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Upper case U represents a full glass
Lower case u represents a half glass
UUuuu UUuuu UUUu
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Upper case U represents a full glass
Lower case u represents a half glass
UUuuu UUuuu UUUu
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Congratulations, Nadir!
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It's interesting you found the other solution. I got this one:
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UUUu UUUu Uuuuuu
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A large bowl has 20 red marbles and 30 blue marbles in it. If you are unable to see what you're doing, and you pick one marble out of the bowl at random, what are the chances you will find the marble you picked is red?
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It seems too simple so I must be wrong! 40% or 0.4.
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A large bowl has 20 red marbles and 30 blue marbles in it. If you are unable to see what you're doing, and you pick one marble out of the bowl at random, what are the chances you will find the marble you picked is red?
It depends on the Vegas odds and how much $ was bet.
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It seems too simple so I must be wrong! 40% or 0.4.
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You got it! Yaaaay!
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Probablility is nothing but the ratio:
Total Target Outcomes
Total of All Possible Outcomes
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In this case, there are a total of 50 marbles you could select at random (there are 50 marbles in the jar).
So that's the denominator.
Then there are 20 red marbles, any one of which is a "target" or a desired selection.
So that's the numerator. Consequently, you simply plug in the numbers and do the arithmetic:
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20
50
= 2/5 = 0.4 = 40%
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Likewise, if you wanted a blue marble, it would be: 30/50 = 3/5 = 0.6 = 60%
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Then to figure for multiple selections, you compute the probability for each draw, and multiply the probabilities together.
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What is the next number in this sequence? 1, 12, 1, 1, 1, ...
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You take one section of a newspaper and pull out one sheet from among them, to find that page 6 and page 25 are printed on the same side of the sheet. From this fact, can you predict what the page numbers are, printed on the center sheet of the same section?
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15,16?
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15,16?
:cheers: ! ;D !
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I think this one is usually harder. You really have to think outside the box!
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What is the next number in this sequence? 1, 12, 1, 1, 1, ...
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It depends on the Vegas odds and how much $ was bet.
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I don't think Las Vegas has any bowls with 30 blue and 20 red marbles in them.
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But if Vegas did have bowls with 30 blue and 20 red marbles, and John bet $5 he would draw a red one, then what?
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You want to boil an egg for exactly 5 minutes, but all you have is a 4-minute timer and a 3-minute timer.
Can you plan how to use these two timers to measure the 5 minute cooking time?
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I think this one is usually harder. You really have to think outside the box!
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What is the next number in this sequence? 1, 12, 1, 1, 1, ...
Any number you like.
Choose any 6th number, and one can construct a rule which predicts the sequence.
Same with respect to cosmology. You can find any number of sets of equations that match astronomical observations. Or in maths, you can find any number of polynomials (or other functions) interpolating a given finite set of points.
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You want to boil an egg for exactly 5 minutes, but all you have is a 4-minute timer and a 3-minute timer.
Can you plan how to use these two timers to measure the 5 minute cooking time?
You operate them in periodical mode. Then you have a beep every
0 4 8 12 16 20 ... minutes
as well as every
0 3 6 9 12 15 18 ... minutes
Five minutes for your egg is e.g. from 3 min (second timer) to 8 min (first timer).
The 3 minutes, before you can start cooking, are not lost. You can e.g. roll a cigarette, light it, and contemplate the quantity of useful plants God created.
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You operate them in periodical mode. Then you have a beep every
0 4 8 12 16 20 ... minutes
as well as every
0 3 6 9 12 15 18 ... minutes
Five minutes for your egg is e.g. from 3 min (second timer) to 8 min (first timer).
The 3 minutes, before you can start cooking, are not lost. You can e.g. roll a cigarette, light it, and contemplate the quantity of useful plants God created.
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Very good, Struthio! You surprised me. Listing the periodical beeps is an interesting development.
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This kind of puzzle is older than electronic timers. The answers given for use of hourglass sand timers can also work.
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(https://s15-us2.startpage.com/cgi-bin/serveimage?url=http%3A%2F%2Fatentif.com%2Fwp-content%2Fuploads%2F2016%2F08%2Fproduct-atentif-3.jpg&sp=fa4b3f3b519531386b2b450e322a31e2)
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The sand timer can be paused in progress by turning it sideways, for example.
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But even though it doesn't make a "beep," by keeping an eye on the timers you can solve the puzzle in a similar way as you did.
This would be the case for let's say a deaf person who can't hear the beep -- he would have to watch any timer.
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1. Start heating the pan with water to boil the egg, but don't put the egg in yet.
2. While waiting for the water to boil, turn both timers over, so the sand starts to flow in each.
3. When the 3-minute timer runs out of sand, immediately turn the 4-minute timer sideways, to pause it in action.
(The image above was selected to show the larger timer turned sideways, but imagine the smaller timer has run out.)
4. When the water begins to boil, place the egg into the water and resume the 4-minute timer, which has one minute of sand left.
5. While the egg boils this first minute, watch the 4-minute timer for it to run out of sand -- when it does, immediately turn it over so as to measure the remaining 4 minutes of egg-cooking time. One minute plus four minutes gives precisely a 5 minute egg.
6. Don't forget to REMOVE the egg from the boiling water as soon as the 4-minute timer runs out of sand.
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Any number you like.
Choose any 6th number, and one can construct a rule which predicts the sequence.
Same with respect to cosmology. You can find any number of sets of equations that match astronomical observations. Or in maths, you can find any number of polynomials (or other functions) interpolating a given finite set of points.
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The puzzle doesn't ask you to construct a rule. I realize that's part of how sequence problems are usually presented, but if they are properly done, they ask you to answer what the rule is such that the next number you provide abides by the same rule. So it's not to your discredit for you to expect this puzzle operates by the same pattern you have seen in the past regarding problems of numerical sequence. Like I said, though, you have to think outside the box for this one.
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In this puzzle, no such requirement is made for construction of any rule.
The puzzle does not say to give examples of several rules for several 6th numbers.
But I noticed that you don't provide any such examples!
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Perhaps you could give just one example of a rule you have successfully constructed and its corresponding 6th number??
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However, for there to be a unique 6th number, some rule must apply.
The puzzle asks what is THE next number (not what could be any of the many next numbers).
And there is a unique answer that fulfills all the requirements.
Therefore, what is the alternative to "constructing a rule?" (big hint!)
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Incidentally, this puzzle could have easily asked for the next 4 numbers in the sequence, but the 6th number is sufficient and unique.
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This is WAY more information than most recipients of this puzzle have had previously.
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It has been reported that the vast majority of people cannot solve it. So don't feel left out if it's too difficult.
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Perhaps you could give just one example of a rule you have successfully constructed and its corresponding 6th number?
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What, for example, would be your "constructed rule" if the chosen 6th number would be 19?
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What, for example, would be your "constructed rule" if the chosen 6th number would be 19?
There is an online Lagrange Interpolation Calculator at easycalculation.com (https://www.easycalculation.com/lagrange-interpolation.php).
In the box labelled "x" you enter 1,2,3,4,5,6
In the box labelled "y" you enter 1,12,1,1,1,19
In the box labelled "Value of corresponding to x" you enter 7
After clicking the "Result" button, the result is shown in the box labelled "P(x)". The result is 175 giving you the 7th number:
1,2,3,4,5,6,7
1,12,1,1,1,19,175
Entering these numbers and "Value of corresponding to x" 8, you get P(x) = 764
1,2,3,4,5,6,7,8
1,12,1,1,1,19,175,764
The formula (i.e. the rule) used is shown below. Mathematicians will make up any number of different formulas you can use to find different curves with different properties "connecting the dots".
Formula used:
(https://www.easycalculation.com/lag-img1.png)(https://www.easycalculation.com/lag-img2.png)(https://www.easycalculation.com/lag-img3.png)
Where, x and y are the coordinates
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That's an interesting website, which would seem to be a reasonable basis for your outlook on this topic.
However, there seems to be a problem.
The formula it provides for the "rule" of operation is always the same formula, regardless of the numbers chosen for the series.
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I tried to change the 6th number from 19 to 13, and the site offered 139 as the 7th, giving the same formula.
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I tried changing the 5th number from 1 to 17 (1,12,1,1,17,...) and the site offered 26 as the 6th, giving the same formula.
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I tried changing the 6th number from 26 to 1 (1,12,1,1,17,1,...) and the site gave -173 as the 7th, giving the same formula.
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No matter what numbers I provide, it guesses the wrong number for the next in the series.
But as soon as I correct the wrong number, it provides another wrong number for the subsequent number following.
And in all cases, the same exact formula is provided as the "rule" that determines the next number.
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So it's not working.
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Try putting in 1,12... and asking for the third number. The site says that 23 is the third number. Wrong: it should be 1.
Try putting in 1,12,1,... and ask for the fourth. It says -32 is the fourth. WRONG: it should be 1.
Try putting in 1,12,1,1,... and ask for the fifth. It says 45 is the fifth. WRONG: it should be 1.
Try putting in 1,12,1,1,1,... and ask for the sixth. It says -54 is the sixth. WRONG: it should be the one I asked for in the puzzle.
We already tried 1,12,1,1,1,19,... and for the next, received 175.
We already tried 1,12,1,1,1,13,... and for the next, got 139.
We already tried 1,12,1,1,1,17,... and for the next, got 26.
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And the real kicker is,
FOR ALL THESE WRONG NEXT NUMBERS AND THEIR CORRECTIONS, THE FORMULA REMAINS THE SAME.
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So just for kicks, let me give you the correct 6th number, which is 2. The site has been unable to provide that number.
Now, given the correct series from the puzzle, 1,12,1,1,1,2,... what does the site give as the formula?
That's right. It provides the same formula as it did for 1,12,1,1,1,... followed by 19, 13 and 17, all of which were WRONG.
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What does the site provide now for the 7th number in the series after 1,12,1,1,1,2,...? It says 73, which is WRONG.
This site, as interesting as it is, has been UNABLE to correctly anticipate ANY of the numbers in this series, even the THIRD (1).
And it gives the same formula FOR EVERYTHING, which by all reasonable expectations, must also be WRONG.
This formula has exactly NOTHING to do with the very simple fact which correctly predicts all the numbers in the puzzle series.
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Formula used:
(https://www.easycalculation.com/lag-img1.png)(https://www.easycalculation.com/lag-img2.png)(https://www.easycalculation.com/lag-img3.png)
Where, x and y are the coordinates
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The site offers calculation of Lagrange polynomials.
Apparently what this puzzle asks for has nothing to do with Lagrange polynomials.
So their formula doesn't work on this puzzle.
If Lagrange polynomials are for constructing curves, perhaps it would help to know that the puzzle is not based on any curve.
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There have been a few people in history who believed that mathematics is the answer for everything.
Some cosmologists today are like that, who think there can be discovered a Theory of Everything, for example.
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But they are quite easily found out to be barking up the wrong tree.
When a simple question like this puzzle can throw a monkey wrench into their whole gearbox, rendering it utterly useless.
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There is no mathematical formula or geometric progression or Lagrange polynomial series that can predict the 6th number.
Or for that matter, the 7th or 8th, either.
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And furthermore, the statement that you can pick any number you want for the 6th, is shown to be false.
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Neil Obstat,
the question I answered was
What, for example, would be your "constructed rule" if the chosen 6th number would be 19?
I presented a rule which can be used to produce the sequence 1,12,1,1,1,19.
Here's another answer, showing that the next number is not the number two:
What is the next number in this sequence? 1, 12, 1, 1, 1, ...
The next number is the number one. The sequence gives the number of cuckoo calls of a cuckoo clock at a quarter to twelve, twelve, a quarter past twelve, half past twelve, a quarter to one, one, etc.
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Neil Obstat,
the question I answered was
I presented a rule which can be used to produce the sequence 1,12,1,1,1,19.
Here's another answer, showing that the next number is not the number two:
The next number is the number one. The sequence gives the number of cuckoo calls of a cuckoo clock at a quarter to twelve, twelve, a quarter past twelve, half past twelve, a quarter to one, one, etc.
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Well, you finally are getting close to the key okay, the number of times a clock chimes, but you missed the answer again.
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Any cuckoo clock that has that sequence is, well, cuckoo. What good is having a call of "one" for 7 quarter-hours in a row?
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Since you're hot on the trail, given that the 6th number is in fact a 2, even if you don't agree, what would be the 8th number?
And don't say it's a 1.
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Try thinking chimes, instead of cuckoo calls.
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BTW I tried the website and it could not predict or produce the number 19 when the 6th position is unkown.
The best it could do was to predict -54 in the 6th position, not 19.
What it did was take 1,12,1,1,1,19 and say that 175 would be the 7th number, but it could not provide 19 for the 6th.
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I went ahead and tried a number of possible continuations of this sequence, leaving out different numbers.
The Lagrange polynomial utility was never able to predict any one of the missing numbers, any which way.
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I input 1,2,4 for x, and 1,12,1 for y. This means our original series with the 3rd number missing: 1,12,...,1
The calculator provided P(x) = 12 which translates to their answer being 1, 12, 12, 1 --- but it should be 1,12,1,1
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Next, I input 1,2,5 for x, and 1,12,1 for y. This would be for our series with the 3rd and 4th numbers missing: 1,12,...,...,1
The calc gave P(3) = 15.67 and P(4) = 12 which means their answer is 1, 12, 15.67, 12, 1 --- should be 1,12,1,1,1
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Next, I input 1,2,6 for x, and 1,12,2 for y. This is for our series with the 3rd, 4th and 5th missing: 1,12,...,...,...,2
The calc gave P(3) = 17.60, P(4) = 17.80, P(5) = 12.60 or, 1, 12, 17.60, 17.80, 12.60, 2 --- should be 1,12,1,1,1,2
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When I left out the first number things got strange.
2,3,5 for x, 12,1,1 for y outputs 30.33, 12, 1, -2.67, 12 --- instead of 1,12,1,1,1,2
2,3,4,6 for x, 12,1,1,2 for y outputs 42, 12, 1, 1, 2, 4
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This formula is utterly useless on this puzzle. The formula clearly has zero relation to the puzzle key.
Sometimes relying on pure mathematics puts you everywhere EXCEPT in the right place.
This polynomial calculator is entirely unable to anticipate any one of the numbers in the subject series!
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By asking for each next number in this sequence beginning with the 6th, given the correct preceding numbers, here is what the calculator provides as a prediction, one position added at a time:
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1, 12, 1, 1, 1, -54, 73, -97, 161, -296, 3217, -1407, 37749025
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Which is absolutely 100% wrong.
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To be clear, I gave the site all the correct sequence numbers up to and including the 12th number and it gave me an answer for the 13th position that is 99.9999974% wrong. IOW 100% wrong.
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Neil Obstat,
what about deaf persons who can't hear the chime? They would have to watch the cuckoo. Don't you remember what you said about the advantages of hourglass sand timers?
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Neil Obstat,
what about deaf persons who can't hear the chime? They would have to watch the cuckoo. Don't you remember what you said about the advantages of hourglass sand timers?
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You have a point there. But maybe you didn't notice that I suggested you stop thinking about cuckoo clocks.
But you keep on anyway. This is not that difficult but if you refuse to pay attention to clues, it gets difficult.
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There are "chiming" clocks that are made for deaf people (flashing lights, etc.), and there are watches made for the blind.
Among those unusual and specialized timepieces, some are of the type that this puzzle applies to.
This puzzle refers to patterns in the greater majority of all timepieces ever made that signal time automatically at select intervals.
You've just been stuck on the wrong interval. (hint, hint)
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As a matter of fact, clocks that chime on the quarter hour most often have a different tone for 15 and 45 minutes after the hour.
Westminster chime for example has 4 different melodies it plays at the 4 quarter hours.
There are a variety of other chimes but the Westminster is the most popular of the musical chimes, in the western world, anyway.
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Westminster chime for example has 4 different melodies it plays at the 4 quarter hours.
There are a variety of other chimes but the Westminster is the most popular of the musical chimes, in the western world, anyway.
With new European Union copyright laws the U.K. will have to pay to continue using Georg Friedrich Händel's original work used by the clock tower at the Palace of Westminster (Westminster Quarters (https://en.wikipedia.org/wiki/Westminster_Quarters#History)). I assume that this will break their economy.
Since you're not a friend of cuckoos, here an alternative from Augsburg:
https://www.youtube.com/watch?v=gfKpSn2JKKo
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I didn't say I don't like cuckoos. I brought a cuckoo clock back from the Black Forest in Germany.
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So if it's cuckoo clocks you prefer to reference, try this one.
This is from a site that sells hundreds of models, and they all have this same function, so perhaps you don't know much about cuckoo clocks! (No mention of 1/4 or 3/4 hour gong strikes for any of their clocks)
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(https://billsclockworks.com/merchandise/images/romba/8223_sm.jpg)
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While the clock cuckoos the hour or half-hour the door opens, a cuckoo appears and calls along with the clock's gong strike. Has the following features: Runs 8 days on one winding, hand carved, musical, animated dancers, made in Germany, wooden hands, numbers, and bird; shut-off switch, official Black Forest clock, licensed by the VDS.
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With new European Union copyright laws the U.K. will have to pay to continue using Georg Friedrich Händel's original work used by the clock tower at the Palace of Westminster (Westminster Quarters (https://en.wikipedia.org/wiki/Westminster_Quarters#History)). I assume that this will break their economy.
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How many ways can you get stuff wrong, Struthio? Handel originally composed the symphony the Westminster Chime was BASED on, but he did not compose the chime. In fact, it is only the four pitches that occur in his symphony at bars 4 and 5 that are used in the Chime, which has 5 variations that comprise its composition. The Westminster Chime was composed by either Regis Professor of music, Dr. John Randall, or his pupil, William Crotch in 1793. The carillon was first played in the bell tower of St. Mary's the Great, the University Church, in Cambridge. Later it was reproduced on a large scale in the Victoria Tower Clock of the House of Parliament (renamed Elizabeth Tower in 2012).
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(http://www.clockguy.com/SiteRelated/SiteGraphics/RefGraphics/ClockTunes/WestminsterTune-HerschedeCatalogue.jpg)
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Speaking of which, here is another puzzle!
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What does how many tones in the Westminster Chime separate the midpoint of the only repetition of the same two pitches from the beginning of the repetition of the entire sequence of tones have in common with the variable pitches themselves?
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And another one:
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If the name of the structure that houses the Great Clock of the House of Parliament, is Victoria Tower, why is it popularly referred to as "Big Ben?"
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And another:
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If the Westminster Chime is rendered in the key of D, what pitch and octave does Big Ben have?
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You are right, Neil Obstat. But the European Union may not act in a reasonable way.
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Neil Obstat,
can you prove that the puzzle in the OP
Can you move just one digit to a new location, making this equation true?
Do not move signs, only move one numeral.
82 - 43 = 1
beside the single solution, found by MaterDominici:
82 - 34 = 1
has no other solutions?
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You are right, Neil Obstat. But the European Union may not act in a reasonable way.
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Are you aware of BREXIT?
Now THERE is a novel explanation, what was truly at the starting point, what sparked Brexit: the Brits were not about to pay duties.
If you put it up for a vote in the UK, the proposal to pay the EU rent on the peal of Big Ben would be an overwhelming loser.
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Are you aware of BREXIT?
Sure. That's why I expect retaliation by the European Union.
what sparked Brexit: the Brits were not about to pay duties.
Sitting in a pub, not being about to pay duties, I would stay for another beer, not leave.
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Neil Obstat,
can you prove that the puzzle in the OP 82 - 43 = 1 [besides] the single solution,
found by MaterDominici: 82 - 34 = 1 has no other solutions?
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If you can provide an alternative solution, that would be wonderful!
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But as for the possibility of proving a negative, well, no, I don't think so. A negative is something that can't be proved.
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There is one reply that gets pretty close: -3.375 = 1 ...but it's still a false equation.
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Or, if you can make the digit you're moving into its reciprocal, you can get closer: 0.399765767... = 1 ...still false.
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Or, by moving a different digit and using its reciprocal you can get closer still: 1.442561476... = 1 ...still false.
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(82)/3 - 4 is pretty close to 8(23) - 40 but you have to leave a 0 where the three came from, and neither is even close to 1.
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Or, if you can introduce a new digit (0) with the excuse that "Zeros are everywhere," then you can have (82-43)0 = 1
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In general, the thing that makes good puzzles so charming is that they have one unique solution that is much simpler than all the runner-up answers. It is the beauty of simplicity that sets them apart. The simpler, the better!
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If the name of the structure that contains the Great Clock of the House of Parliament is "Clock Tower" (or "Victoria Tower" renamed Elizabeth Tower in 2012), then why is it popularly referred to as "Big Ben?"
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A simple question with a simple answer.
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The movement and technical description of the various parts of this historic clock are a topic of vast interest by clockwork enthusiasts worldwide. Some of the numbers and materials describing its parts and operation are frankly hard to believe, when considered out of context. The fact that it's been in service under all weather conditions for 159 years and continues to operate reliably is quite a testament to its designers and craftsmen who made it a reality. It survived the onslaught of German bombs in WWII, with the peals of clock chimes not missing a beat for the duration even amidst the explosion of ordnance. There have been lists of trivia facts drawn up about this "Great Clock" (a moniker in use for over a century already) that are quite amusing -- especially once you know a thing or two about pendulum clocks. Yes, it is regulated by a pendulum, with a period of two seconds; 13 feet tall, weighing 660 pounds, and costing a lot more! Incredible, in an age of electronic everything. It is 100% mechanical, a purist's dream come true. It gets wound three times a week by hand, a process that takes 90 minutes. For the English (thank goodness) it would be UNTHINKABLE to replace this enormous movement with a more modern one that would work automatically, perhaps by wind power. Some have said that they were a little short-sighted by not making the actual pendulum VISIBLE through a window of some kind on the sides of the tower. The regular 2-second beat of this clock's heart could be made to appear as a sparkling Jєωel in the center of the tower. In answer to this charge, it has been countered with the fact that this is a FOUR-SIDED CLOCK and its pendulum operates in only two directions, so to make it visible, only two of the four sides would be able to display its movement. Well, that's not a closed issue. By use of prisms or mirrors, the image of the pendulum in motion could conceivably be conducted to appear similarly through windows in each of the four sides. The cost of making such changes do not qualify as urgent, however, when compared to paying for essential maintenance. Certainly, four TV screens displaying the PICTURE of the pendulum in motion could be hung on each of the 4 sides, but that would not be in keeping with the longstanding character of the clock: all mechanical, no electricity required. The whole ambiance of the Great Clock is a cultural icon of the British people. In 2007 it was found to be the single most popular tourist attraction and thought to be the single most culturally relevant item on English soil. IOW Big Ben is as British as motherhood and apple pie is American. They've just managed to turn their national symbol into a tourist attraction, unlike Americans. But having their "pie" and eating it too comes with a price! As of 2017, necessary maintenance and functional renovations are in progress which are estimated to cost £61 million. During this repair process the chimes will be silent for several months. We should hope this will not stir up civil unrest.
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Big Ben's the bell innit
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Big Ben's the bell innit
Do you mean 'Big Ben is the bell in it.' or 'Big Ben is the bell, innit?'?
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Do you mean 'Big Ben is the bell in it.' or 'Big Ben is the bell, innit?'?
Both.
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Let me guess -- innit as in, "That's the answer, innit?" (That's the answer, isn't it?")
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Answer: Big Ben is the name of the bell that chimes the hour count. It weighs 13.5 tons. And it's cracked. If it were not cracked it would sound even better. BTW the Liberty Bell is cracked too, but the difference is, Americans don't use it to hear its peal.
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Big Ben is alternately used referring the bell itself, the Great Clock, or the tower in which they're located.
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If the name of the structure that contains the Great Clock of the House of Parliament is "Clock Tower" (or Victoria Tower renamed Elizabeth Tower in 2012), then why is it popularly referred to as "Big Ben?"
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Big Ben is alternately used referring the bell itself, the Great Clock, or the tower in which they're located.
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When Big Ben rings the number of times equal to the current hour, it's after 4 other (smaller) bells have all played their 20-tone sequence that precedes the first strike of Big Ben on each hour. The clock is maintained with the objective of having the first strike of the hour occur within one second of the hour, UTC (erstwhile Greenwich Mean Time or GMT). IOW bankers should be able to set their watches by the chime of this clock. Maintenance men can slow the pendulum of Big Ben one second per hour by adding a British coin to a particular stack of coins held in place on the pendulum, or, they can speed it up by removing one coin. On the prelude to each hour chime of Big Ben, the two highest pitched bells each play their notes 5 times, while the lowest plays its note just 4 times and the second lowest of the 4 plays its note 6 times.
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Listeners on the far side of London hearing the chime of Big Ben both by radio reception AND live through the air, can hear the 12 o'clock peals 13 times. The first peal is via the radio alone, followed by 11 peals of the radio in unison with the live sound through the air, and the 13th peal is only via the air (not on the radio). These people are located far enough away from the House of Parliament such that the time it takes for the sound of Big Ben to reach them equals the time interval that separates each peal of Big Ben, about 4.5 seconds. Therefore, they hear the second radio peal while they're hearing the first live peal, etc.
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Radio ... Live
1
2 . . . 1
3 2
4 3
5 4
6 5
7 6
8 7
9 8
10 9
11 10
12 . . 11
12
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Here's one everybody can enjoy!
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TEA WITH MILK
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Given two 8-ounce glasses, one half full of milk, the other half full of tea; what do you end up with, after the following:
Take a teaspoon of milk out of the milk glass and stir it into the glass with tea in it.
Then take a teaspoon of the tea-milk mixture and stir it into the glass with the milk in it.
Is there more milk in the tea than tea in the milk, or more tea in the milk than milk in the tea?
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Here's one everybody can enjoy!
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TEA WITH MILK
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Given two 8-ounce glasses, one half full of milk, the other half full of tea; what do you end up with, after the following:
Take a teaspoon of milk out of the milk glass and stir it into the glass with tea in it.
Then take a teaspoon of the tea-milk mixture and stir it into the glass with the milk in it.
Is there more milk in the tea than tea in the milk, or more tea in the milk than milk in the tea?
There is more milk in the tea (1 teaspoon of milk was stirred into 4 ounces of tea) than tea in the milk (1 teaspoon of milk-tea mixture was stirred into the 4 ounces of milk. Thus only 1/2 teaspoon of tea + 1/2 teaspoon of milk was put into the 4 ounces of milk)
But one could also say that the 4 ounces of milk contains less tea (only 1/2 teaspoon of tea) and more milk.
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There is more milk in the tea (1 teaspoon of milk was stirred into 4 ounces of tea) than tea in the milk (1 teaspoon of milk-tea mixture was stirred into the 4 ounces of milk. Thus only 1/2 teaspoon of tea + 1/2 teaspoon of milk was put into the 4 ounces of milk)
But one could also say that the 4 ounces of milk contains less tea (only 1/2 teaspoon of tea) and more milk.
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The underlined words above are correct (sort of). The rest, unfortunately, are incorrect.
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The puzzle does not say 1/2 tsp of tea + 1/2 tsp of milk, that is your mistaken deduction.
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The contents of the second teaspoon is about 99% tea and only about 1% milk.
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To be precise, the second teaspoon of mixture was stirred into the 3.99 oz of milk (approximately).
Remember, a tsp of milk had been removed, so it was less than 4 oz.
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The second teaspoon contained 99% tea and 1% milk in it (approximately). It was not half tea and half milk!!
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These approximations are okay because the two cups contained the same volumes: one milk, one tea, each was 4 ounces.
And the same volume was removed from the milk as was removed from the tea-with-milk mixture: 1 tsp.
One tsp 100% milk, followed by one (99% tea + 1% milk) tsp (approximately).
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Consequently, your second statement, "1 teaspoon of milk-tea mixture was stirred into the 4 ounces of milk" should be amended to say, 1 teaspoon of milk-tea mixture was stirred into the (4 ounces minus 1 tsp) of milk.
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It went like this:
(For simplicity's sake, presume 1 tsp = 1/100th of the 4-oz volume)
1 tsp milk removed from milk glass, added to tea glass =
tea glass = 100 parts tea + 1 part milk; milk glass = 100 parts milk - 1 part milk = 99 parts milk
Volume of tea glass = 101 parts Volume of milk glass = 99 parts
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1 tsp tea/milk mixture (99% tea, 1% milk) removed from tea glass, added to milk glass =
tea glass = 99.0099... parts tea + 0.990099... part milk; milk glass = 99.0099... parts milk + 0.990099... part tea
Simplified: (99.01 parts tea + 0.99 part milk) in tea glass; (99.01 parts milk + 0.99 part tea) in milk glass
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Bottom Line - you end up with:
1 glass with 99% tea 1% milk, 1 glass with 1% tea 99% milk ---- IOW the same milk-in-tea, as tea-in-milk.
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The number of ounces in the glasses is irrelevant, so long as they are the same. (The original volume is 4 ounces each.)
They could be 20 oz or 3,419 ounce glasses, no difference. So long as they're the same volume.
The puzzle says, "8 ounce glasses... half full..." just to distract you and to make you think of half volumes.
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The total volume in each glass, regardless of its size, is unchanged by the transfer.
The same volume is removed, transferred, removed and transferred back.
So the first transfered volume is canceled by the second.
And the volume of milk transferred in the first teaspoon is exactly canceled with the volume of tea/milk transferred back.
If this process were repeated many times, eventually both glasses would end up with 50/50 tea and milk, with the same volume.
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The underlined words above are correct (sort of). The rest, unfortunately, are incorrect.
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The puzzle does not say 1/2 tsp of tea + 1/2 tsp of milk, that is your mistaken deduction.
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The contents of the second teaspoon is about 99% tea and only about 1% milk.
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To be precise, the second teaspoon of mixture was stirred into the 3.99 oz of milk (approximately).
Remember, a tsp of milk had been removed, so it was less than 4 oz.
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The second teaspoon contained 99% tea and 1% milk in it (approximately). It was not half tea and half milk!!
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These approximations are okay because the two cups contained the same volumes: one milk, one tea, each was 4 ounces.
And the same volume was removed from the milk as was removed from the tea-with-milk mixture: 1 tsp.
One tsp 100% milk, followed by one (99% tea + 1% milk) tsp (approximately).
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Consequently, your second statement, "1 teaspoon of milk-tea mixture was stirred into the 4 ounces of milk" should be amended to say, 1 teaspoon of milk-tea mixture was stirred into the (4 ounces minus 1 tsp) of milk.
Okay, thanks. It is late. I see the mistake I made.
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So far there are two puzzles in this thread nobody has tried to answer:
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What does how many tones in the Westminster Chime separate the midpoint of the only repetition of the same two pitches from the beginning of the repetition of the entire sequence of tones have in common with the variable pitches themselves?
(Hint: evaluate the two phrases separately: 1) the beginning of the repetition of the sequence, and 2) the midpoint of the only repetition of two pitches.)
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And another:
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If the Westminster Chime is rendered in the key of D, what pitch and octave does Big Ben have?
(Hint: the image I provided of the music is transposed to D, even though Big Ben does not sound in that key.)
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If the Westminster Chime is rendered in the key of D, what pitch and octave does Big Ben have?
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Big Ben is the name of the bell that chimes the number of hours at the top of the hour, 24 times each day.
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Rendered in the key of D, Big Ben has the pitch of B and is one octave below the high B in the 4-pitch bells.
If translated to sol-fas, the final pitch would be La.
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The previous post I made has this transposition written out (the low B is under the word "Hour.") :
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(http://www.clockguy.com/SiteRelated/SiteGraphics/RefGraphics/ClockTunes/WestminsterTune-HerschedeCatalogue.jpg)