Your readers have found a way to boil an egg for 2 1/4 minutes:
8,0; 4,4; 2,6; 1,7; 4.5,3.5; 2.25,5.75
But it is possible to get your soft-boiled egg 4.5 minutes earlier:
8,0; 4,4; 2,6; 5,3; 6.5,1.5
put egg in boiling water, measure 1.5 minutes, then immediately tilt again
for 0.75 more minutes.
With two pyramid egg timers it's possible to reach 2.25 minutes even
quicker- 8 minutes quicker I believe.
Regards
Jonathan
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I tried to do the same for the tetrahedral, but it seemed to lack all of the patterns and symmetry of the triangular one. It basically came out to be a big mess.
Unlike the triangular hourglass, the tetrahedral cannot (I think) make every amount.
I started with 27 minutes in the first bulb and after 4 turns my tree starts repeating some numbers. For example, you can make 1/3, 4/3 and 7/3 but not 10/3 and you can make 17+1/3 in 2 different ways.
-Jeremy Galvagni
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I made a neat picture using geometers sketchpad showing all possiblities for up to five turns.
The dark line is the solution for 2.25 minutes. Unfoutunately, the directions of the segments doesn't correspond to the way the hourglass should be turned (LLLRL).
But it does show an interesting branching.
Jeremy Galvagni
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Dear Ed,
Thank you for publishing the Hourglas at www.mathpuzzle.com.
I think I have found the general algorithm for setting the Binary
(three-legged) Hourglas.
1) Write the desired time interval in binary digits as accurately as you
need. E.g. two-and-a-quarter minutes = 10.01 binary minutes. (I guess that
there is a better way of describing this step mathematically)
2) Make the variable x equal to this time interval.
3) Note down x on paper.
4) If x is larger than 4, then take the 8-complement (i.e x --> 8-x)
5) Double the interval (i.e. x --> 2x)
6) If x<>8 then goto 3)
7) Reverse the list of x-es that you have noted down in the repetitive steps
3)
8) Set your Hourglas according to this reversed list
I have not been able to find any logic in the Pyramid (four-legged)
Hourglas. Examining the achievable split ratios in a few steps, I could not
find any pattern.
Best regards,
M. Oskar van Deventer
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To measure 2.25 minutes with the 3-sided hourglass, follow the example
to reach the state with 4.5 and 3.5 minutes of sand in two of the
chambers, then bring the 4.5 minute side up. 2.25 minutes worth of sand
will flow into each of the other sides, one of which was previously
empty, so it now contains 2.25 minutes worth of sand.
In the general case, you can work out the manipulations needed to
measure any given amount of sand by working backwards. In each reverse
step, the amount of sand in the side with less sand, plus the same
amount from the other side, flows back into the third chamber, thus
doubling that smaller amount. Keep going and you will eventually reach
the 2 miunte-6 minute state which all preparation phases must go through
unless you want to measure 8 or 4 minutes.
You have a broken image for "Oskar's Pyramid" because it is looking for
the image on your C: drive. But it seems to me that the 3-sided
hourglass is more useful than the 4-sided one, unless you like counting
time in trinary fractions.
You can represent the preparation phase of the 3-sided hourglass in
terms of a binary string. If 1 represents turning up the side with more
sand, and 0 the side with less sand, then the first two steps are always
10 or 11 (equivalent; I will use 11). Then 2 minutes is 110. 6 minutes
is 111. 7 minutes is 1101. 1 minute is 1100. 2.25 minutes is 110110. We
don't strictly need the first two symbols if you don't mind not having a
way of writing 4 or 8 minutes.
This doesn't seem to correspond with binary numbers. Does it correspond
with some other known code, after an appropriate manipulation?
Joseph DeVincentis
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Right-Left-Right-Right-Right-Left
8-0
4-4
6-2
7-1
4.5-3.5
2.25-5.75
Regards,
-Jared Marks
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Oskar has already separated the sand into 4.5 minutes and 3.5 minutes.
Instead of turning it left as he does to time 3.5 minutes, he should turn it
right, thereby splitting the sand into 2.25 minutes and 5.75 minutes. Then
he can time his 2.25 minute egg.
Luke Pebody
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Hi Ed,
Using an 8-minute, 3-legged hourglass one can measure
2.25 minutes by preparing with right-left-right-right-right,
to give 2.25 in the right leg. Turn left to measure.
I found it by working backwards. There are two ways to
get to (A,B):
-turn rigth from (2B,A-B)
-turn left from (B-A,2A)
but since A+B=8, either 2A or 2B must be >8 (except for (4,4)),
so only one of them can be valid.
Thus, you can repeatedly apply this method to find the solution,
which will be the only one.
Note:
In theory it is possible to achieve any time interval between
0 and 8 minutes. However, in practice any number that
does not have a finite number of digits when written in binary
(i.e. is not a sum of integer powers of 2), will require an
infinite number of turns in the preparation phase.
The backwards-working algoritm above will only succed with
numbers that have a finite solution.
The only way to make (2.25, 5.75) is to turn (4.5, 3.5) right,
and that's where the given example stops.
Thanks,
Trygve Flathen
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Hello,
To get the three-legged Hourglass to boil an egg exactly 2.25 minutes use the following pattern.
right, left, right, right, right
Please let me know if this works.
Johan Maatje
Lethbridge, Alberta
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Hi Edd,
I have the solution for the 3-sided Hourglass, I can't belive that a puzzle
in your site is so simply, maybe I didn't understand it very well (it's
possible 'couse my English is very bad, as you can see). The solution is yet
in your site, starting from the solution with 4.5-3.5 (the last one) we have
to turn the hourglass clockwise and we had 2.25-5.75 that is the solution we
were looking for.
It's all right?
Best regards,
Gabriele Carelli
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Actually, the solution appears already on your website, almost, as 4.5 is twice 2.25.
Thus, beginning with an 8-minute three-sided glass: turn right-left-right-right-right => (5.75 / 2.25). Then, a left turn times the egg.
George Tolley
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Hi,
You *do* realize that the example you gave provides the answer: simply flip
the last hourglass the other way!
--
Pierre Baillargeon
ArtQuest
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Your solution for measuring 3.5 minutes with the three-legged timer is
just one step away from measuring 2.25 minutes: just turn it to the right.
I believe I'll enjoy the Fractal Maze.
See ya,
Bryce
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Ed,
One more turn to the right on the diagram you’ve posted for 3.5 will give you 2.25. Also, every valid time (all of them, if the theory holds true) has a unique solution, since each time configuration has only one possible previous configuration; the bulb that was emptied last will have contained twice the amount of sand that is in the lesser of the two currently occupied bulbs.
Clint Weaver
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"How to use the Three-legged Hourglass to boil an egg exactly 2.25 minutes?"
From the starting position of 0 at top, 8 in lower left, 0 in lower right,
rotate as follows:
Clockwise, clockwise, counterclockwise, counterclockwise, counterclockwise.
Ending position is 2.25 minutes in lower left hourglass.
Seth Kromholz
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HI there...
the answer would be to follow this combination... first we begin with 8 in one and 0 in the others...Then...turn...right..left...right..right...right...
thus we get 2.25 in one side and 5.75 in the other...
Now we can happily boil the proverbial egg in 2.25 mins.
Lance T. Philip
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