Brain Teaser No 1
Posted by: Don Atkinson on 16 November 2001
THE EXPLORER
An explorer set off on a journey. He walked a mile south, a mile east and a mile north. At this point he was back at his start. Where on earth was his starting point? OK, other than the North Pole, which is pretty obvious, where else could he have started this journey?
Cheers
Don
Posted on: 14 February 2003 by Don Atkinson
Matthew,
Thanks for the hints.
288 balls, in equal, alternating layers that touch all six faces of the box (including the lid) and shaped up in the ever popular FCC (well almost FCC) format, rather than the more natural HCP format.
Great.
Cheers
Don
Thanks for the hints.
288 balls, in equal, alternating layers that touch all six faces of the box (including the lid) and shaped up in the ever popular FCC (well almost FCC) format, rather than the more natural HCP format.
Great.
Cheers
Don
Posted on: 14 February 2003 by Don Atkinson
OTOH,
288 divides nicely by 6 to give 48
So we could be looking at six layers of balls
48 balls in a layer looks good when arranged in a 4x12 matrix
with the alternating rows of 4 offset (see picture)
This seems to allow 12 such rows to be fitted comfortably into a 10cm length
In fact there would be a little bit of free space so the 12 rows could be slightly spaced out
This might allow each succeeding layer to settle slightly lower in the box and give room for 6 alternating layers.
Now, I just need to do a little bit of elementary geometry to see if it all fits together!!!.....
Cheers
Don
288 divides nicely by 6 to give 48
So we could be looking at six layers of balls
48 balls in a layer looks good when arranged in a 4x12 matrix
with the alternating rows of 4 offset (see picture)
This seems to allow 12 such rows to be fitted comfortably into a 10cm length
In fact there would be a little bit of free space so the 12 rows could be slightly spaced out
This might allow each succeeding layer to settle slightly lower in the box and give room for 6 alternating layers.
Now, I just need to do a little bit of elementary geometry to see if it all fits together!!!.....
Cheers
Don
Posted on: 14 February 2003 by Don Atkinson
OOOppps
"This seems to allow 12 such rows to be fitted comfortably into a 10cm length
In fact there would be a little bit of free space so the 12 rows could be slightly spaced out...........
Now, I just need to do a little bit of elementary geometry to see if it all fits together!!!....."
Well, a preliminary calculation suggests that the 4x12 matrix shown above, doesn't quite fit into a 10x5 area. It seems to need 10.03x5.00....
Oh well, back to the drawing board....
Cheers
Don
"This seems to allow 12 such rows to be fitted comfortably into a 10cm length
In fact there would be a little bit of free space so the 12 rows could be slightly spaced out...........
Now, I just need to do a little bit of elementary geometry to see if it all fits together!!!....."
Well, a preliminary calculation suggests that the 4x12 matrix shown above, doesn't quite fit into a 10x5 area. It seems to need 10.03x5.00....
Oh well, back to the drawing board....
Cheers
Don
Posted on: 15 February 2003 by Don Atkinson
Bulging Layers....
In my 4x12 matrix c/c dimensions of the 4 balls across the box are 4/7 = 0.5714 and the c/c dimensions of the alternating red/blue rows are (sqrt 33)/7 = 0.8207.
Subsequent layers start with alternating rows of red/blue balls
Now, I had visualised each layer could 'bulge' in the centre or at the ends. In other words, instead of being flat, each layer of balls would be slightly curved in one plane. Just sufficient to reduce to overall length from 10.027 down to 10.0.
good idea.
But, as i said before, there always is a but, the vertical interval of succssive layers is also =0.8207 and 6 of these layers is too thick to fit into the box.
back to the drawing board.
Cheers
Don
In my 4x12 matrix c/c dimensions of the 4 balls across the box are 4/7 = 0.5714 and the c/c dimensions of the alternating red/blue rows are (sqrt 33)/7 = 0.8207.
Subsequent layers start with alternating rows of red/blue balls
Now, I had visualised each layer could 'bulge' in the centre or at the ends. In other words, instead of being flat, each layer of balls would be slightly curved in one plane. Just sufficient to reduce to overall length from 10.027 down to 10.0.
good idea.
But, as i said before, there always is a but, the vertical interval of succssive layers is also =0.8207 and 6 of these layers is too thick to fit into the box.
back to the drawing board.
Cheers
Don
Posted on: 17 February 2003 by Matthew T
In the mean time...
Life is changing in the naim factory with new layouts being investigated in the production area. The process of building the all new CDXXX is one that is complex and after one of the processes adhesives need time to cure and therefore the player can't be moved.
So the plan is split up the square work area into sections allocated to the various workers, rather then have them all working together. The area given to each would be related to the number of CD players that would be waiting for the adhesive to cure.
Of the five workers Richard has to work out the layout and knows that it takes 6 minutes for himself and the three Dougs to do one of the CD players and the speedy Anna managed to get that down to 5 minutes when she stepped in for Richard. The Dougs all work at the same rate and the area could be split into identical square sections, only one of which Richard was left to work in.
How sections where left for the others?
Life is changing in the naim factory with new layouts being investigated in the production area. The process of building the all new CDXXX is one that is complex and after one of the processes adhesives need time to cure and therefore the player can't be moved.
So the plan is split up the square work area into sections allocated to the various workers, rather then have them all working together. The area given to each would be related to the number of CD players that would be waiting for the adhesive to cure.
Of the five workers Richard has to work out the layout and knows that it takes 6 minutes for himself and the three Dougs to do one of the CD players and the speedy Anna managed to get that down to 5 minutes when she stepped in for Richard. The Dougs all work at the same rate and the area could be split into identical square sections, only one of which Richard was left to work in.
How sections where left for the others?
Posted on: 17 February 2003 by steved
Matthew T,
Sorry, you've lost me completely. I think one of us has been drinking!!
Steve D
Sorry, you've lost me completely. I think one of us has been drinking!!
Steve D
Posted on: 17 February 2003 by Dobbin
42 
Posted on: 17 February 2003 by Don Atkinson
Matthew T,
Sorry, you've lost me completely. I know that I haven't been drinking!!.....Yet.
Dobbin,
How much have you had to drink to even think about Matthew's problem. Even if you've got the right answer, it doesn't count!!!
Cheers
Don
Sorry, you've lost me completely. I know that I haven't been drinking!!.....Yet.
Dobbin,
How much have you had to drink to even think about Matthew's problem. Even if you've got the right answer, it doesn't count!!!
Cheers
Don
Posted on: 18 February 2003 by Matthew T
42 is not correct.
No drinking, at least not yet.
Productivity is proportioanl to work area.
Matthew
No drinking, at least not yet.
Productivity is proportioanl to work area.
Matthew
Posted on: 18 February 2003 by Dobbin
42 must be correct - it's the answer to everything else!
Background:
42 (base 13) is equal to 54 (base 10). (Of Course -- Douglas Adams has been quoted as saying " You just don't write jokes in base 13!" )
To dispel any myths about 42 (and to make Douglas Adams Happy!), Douglas Adams also wrote on USENET: The answer to this is very simple. It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one. Binary representations, base thirteen, Tibetan monks are all complete nonsense. I sat at my desk, stared into the garden and thought '42 will do' I typed it out. End of story
Meaningless nonsense?:
The Earth is 42 orders of magnitude (based on powers of ten) larger than the Planck Length (in case you dont know what that is, it is the smallest length that humans have been able to measure; all below the Planck Length is impossible to measure because time and space merge, and relativity ceases to be the rule. The Planck Length is 1 x 10^-35m.)
The angle at which light reflects off of water to create a rainbow is 42 degrees.
A 42 mile per hour wind brought down the original Tacoma bridge in the state of Washington in the 1930's
Convert the letters of the word "Iron sulfide" into numbers of the alphabet. (Iron sulfide being (for the record) the true name of fools gold.) I.e:
I R O N S U L F I D E
9 18 15 14 19 21 12 6 9 4 5
Now then, Adding up the first 3 digits yields a total of 42. Adding up the next 3 digits makes 54, which is what 42 would be in base 10, if 42 was originally written as base 13 (The significance of which is widely known). Adding up the rest gets you 36, which is 42 written in base 12.
An interesting and oft-cited factoid states that when asked to choose a random number between 1 and 20, 17 is the number most often chosen. A quick examination reveals the following:
42 = 7 * 2 * 3
17 = 7 * 2 + 3
"The Sri Yantra is drawn from nine triangles, four pointed downward and five pointed upward, thus forming 42 (6x7)[sic] triangular fragments around a central triangle. There is probably no other set of triangles which interlock with such integral perfection." from "Sacred Geometry" Lawlor R., Thames and Hudson NY,1982 p.9.
42 is the natural vibration frequency of human DNA
42 is the natural vibration frequency of white mouse DNA.
It is 4.2 light-years to the nearest star from our solar-system?
The chamber in the Cheops pyramid is exactly 42 metres over the ground
The total number of dots on a pair of dice is 42
42 is 4 tens and 2 ones. 4 is the only perfect number (the number is a perfect square AND it is the only number to have the same number of letters as the number represents) 2 squared is 4, the perfect number.
Take ANY number, square its digits, and then add them up. Do the same thing with the answer. If you keep doing this indefinitely your final answer will either be a 1, or you'll enter a loop of solutions that -culminate- with the number 42
The Department of Mathematics at the Muenster-University are most interested in the relation between the numbers 42 and 17. Here some examples: - the 17th prime number is 59, and this is the sum of 17 and 42. - If dig(n) means the sum of the digits of n, there are numbers with dig(n)+(dig(n)+1)=n (we call them 'additive dig-numbers') and there are numbers with dig(n)*(dig(n)+1)=n (the so called 'multiplicative dig-numbers') You can show that there is only one 'additive dig-number': 17=(1+7)+((1+7)+1) You can also show that there are only four 'multiplicative dig-numbers': 12=(1+2)*((1+2)+1) 42=(4+2)*((4+2)+1) 90=(9+0)*((9+0)+1) 156=(1+5+6)*((1+5+6)+1) And you see: 42 is one of them.
42 is the number of the beast 6*6+6=42
The latest evolutionary "Missing Link" is 4.2 million years old
Water is most dense at 4.2 degrees celsius. Ice therefore floats on water, and allows marine life to survive below it.
The integer part of the square root of proton mass divided by electron mass gives 42
The element Molybdenum is vital for plant life on earth. The Atomic Number of Molybdenum is 42
5+4 (The parts from 54) = 9; 4+2 (The parts from 42) = 6 "What do you get if you multiply 6 by 9" The answer is all yours.
1932 Is the year when Ford and Arthur return to the present in "Life, the Universe..." and it's also the day that the Vogons came to the earth. Therefore: 1986 (The year of the Copyright) - 54 = 1932!!!
From Nicholas E. Scripture: 50 Mathematical Puzzles and Oddities, Faber and Faber, London,1963, p.47, chapter "Oddments in Algebra", No. 30:
The problem is:
Prove that for any value of 'n', as long as it is a whole
number, (n^7-n) must always be a multiple of 42.
4^2 = 2^4. Other than 24, this is the only non-palindromic two-digit number for which that is true
if you assign the alphabetical value of to the letters of the words: BIG BANG (or GNAB GIB) you get 2+9+7+2+1+14+7 which equals to, indeed, 42
They've found a star that has a planet circling around it for the first time. Distance to this star: 42 light-years
A grad student at UVic says that the lifetime of the universe will be 10^42 years. Note that this is the lifetime, not the age...
Propane boils at -42 C
The maximum temperature in degrees celsius a human can survive in case of a disease is (of course) 42 degrees celsius !
The REAL Hubble is (d / s) distance over speed of galaxies. As we cannot measure the speed accurately, the Constant is not known but scientists think it is betwixt 100 and 40, therefore it could be FORTY-TWO.
If you take the number 'PI' as 3.14, and multiply the whole part (3) by the decimal part (14), 3 X 14 --> 42
The Specific heat capacity of water is 4200 J/kg oC(Joules/Kilograms x degrees celsius)
If you drilled a hole through the Earth from one point to another, and that hole was frictionless, it would take 42 minutes for gravity to move you from you side of the tunnel to the other. That also goes for satellites orbiting the Earth with only gravity moving them.
The galaxy M84 (Messier Catalog) is 42 x 10^6 light years distant. It is an elliptical galaxy in Virgo with a diameter of 25,000 light years
Two physical constants in the universe are the speed of light and thediameter of a proton. It takes light 10 to the minus 42nd power seconds tocross the diameter of a proton.
The Earth is sometimes referred to as the third stone (or rock, if you're a Jimi Hendrix fan) from the sun. A stone (British measurement system) is 14 pounds, so we have 3 x 14 = 42
The world record in memorizing PI is quoted, from the Chicago Sun-Time, to be "over 42,000"
All dogs have a total of 42 teeth over their lifetimes
The average dormancy of an active volcano is 42 years.
The fastest speed for a land animal for 1 mile is 42mph (Guiness, page 59).
Greatest recorded distance covered by one jump of a kangaroo is 42 feet (Guiness, page 71).
The longest lava flow recorded is 42 miles (Guiness, page 129).
The worlds widest conveyor belt is 42 feet wide (Guiness, page 335).
The largest shoes ever sold were size 42 (Guiness, page 361).
A barrel contains 42 gallons.
The time it took for one person to eat an entire ox was 42 hours (Guiness, page 492)
42 occurs infinitely many times in the number PI.
When PI is rounded to the third decimal place contains the number 42
Consider the following: ( ( 1 * 2 * 3 ) - 4 + 5 ) * 6 = 42
In reverse polish notation (postfix), used on most HP calculators (and others), the expression
2 4 2 4 2 * + * +
evaluates to 42! (This, of course, is closely related to the fact that the binary number system, 101010 means 42, or that in base 4, 222 means 42).
SQRT(41*42+SQRT(41*42+SQRT(41*42+SQRT(...))))=42 where SQRT() means the 'square root' of the bracketed expression.
The Efficiency of human respiration is 42
The number of chromosomes of harvested wheat is 42
In the "Tree Information" sketch for Monty Python, the Larch (the one they show over and over again) is tree number 42.
42 in base 4 (222) is the same as in base -5 ie. 54
1 joule is equal to 4.2 calories
The following question appears in the Rhind Papyrus, which is the oldest mathematical book known and originated in Egypt in 1650 B.C.: "If a certain number, two-thirds of it, half of it, and a seventh of it are added together, the result is 97." The number, of course, is 42. (x + 2x/3 + x/2 + x/7 = 97 --> x = 42) This was derived from an 8th grade Addison-Wesley Algebra book, page 444.
42 in base 4 is 222 or 4;222. Also, if you multiply or add the digits together, you get the same thing for 42 and 222.
Background:
42 (base 13) is equal to 54 (base 10). (Of Course -- Douglas Adams has been quoted as saying " You just don't write jokes in base 13!" )
To dispel any myths about 42 (and to make Douglas Adams Happy!), Douglas Adams also wrote on USENET: The answer to this is very simple. It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one. Binary representations, base thirteen, Tibetan monks are all complete nonsense. I sat at my desk, stared into the garden and thought '42 will do' I typed it out. End of story
Meaningless nonsense?:
The Earth is 42 orders of magnitude (based on powers of ten) larger than the Planck Length (in case you dont know what that is, it is the smallest length that humans have been able to measure; all below the Planck Length is impossible to measure because time and space merge, and relativity ceases to be the rule. The Planck Length is 1 x 10^-35m.)
The angle at which light reflects off of water to create a rainbow is 42 degrees.
A 42 mile per hour wind brought down the original Tacoma bridge in the state of Washington in the 1930's
Convert the letters of the word "Iron sulfide" into numbers of the alphabet. (Iron sulfide being (for the record) the true name of fools gold.) I.e:
I R O N S U L F I D E
9 18 15 14 19 21 12 6 9 4 5
Now then, Adding up the first 3 digits yields a total of 42. Adding up the next 3 digits makes 54, which is what 42 would be in base 10, if 42 was originally written as base 13 (The significance of which is widely known). Adding up the rest gets you 36, which is 42 written in base 12.
An interesting and oft-cited factoid states that when asked to choose a random number between 1 and 20, 17 is the number most often chosen. A quick examination reveals the following:
42 = 7 * 2 * 3
17 = 7 * 2 + 3
"The Sri Yantra is drawn from nine triangles, four pointed downward and five pointed upward, thus forming 42 (6x7)[sic] triangular fragments around a central triangle. There is probably no other set of triangles which interlock with such integral perfection." from "Sacred Geometry" Lawlor R., Thames and Hudson NY,1982 p.9.
42 is the natural vibration frequency of human DNA
42 is the natural vibration frequency of white mouse DNA.
It is 4.2 light-years to the nearest star from our solar-system?
The chamber in the Cheops pyramid is exactly 42 metres over the ground
The total number of dots on a pair of dice is 42
42 is 4 tens and 2 ones. 4 is the only perfect number (the number is a perfect square AND it is the only number to have the same number of letters as the number represents) 2 squared is 4, the perfect number.
Take ANY number, square its digits, and then add them up. Do the same thing with the answer. If you keep doing this indefinitely your final answer will either be a 1, or you'll enter a loop of solutions that -culminate- with the number 42
The Department of Mathematics at the Muenster-University are most interested in the relation between the numbers 42 and 17. Here some examples: - the 17th prime number is 59, and this is the sum of 17 and 42. - If dig(n) means the sum of the digits of n, there are numbers with dig(n)+(dig(n)+1)=n (we call them 'additive dig-numbers') and there are numbers with dig(n)*(dig(n)+1)=n (the so called 'multiplicative dig-numbers') You can show that there is only one 'additive dig-number': 17=(1+7)+((1+7)+1) You can also show that there are only four 'multiplicative dig-numbers': 12=(1+2)*((1+2)+1) 42=(4+2)*((4+2)+1) 90=(9+0)*((9+0)+1) 156=(1+5+6)*((1+5+6)+1) And you see: 42 is one of them.
42 is the number of the beast 6*6+6=42
The latest evolutionary "Missing Link" is 4.2 million years old
Water is most dense at 4.2 degrees celsius. Ice therefore floats on water, and allows marine life to survive below it.
The integer part of the square root of proton mass divided by electron mass gives 42
The element Molybdenum is vital for plant life on earth. The Atomic Number of Molybdenum is 42
5+4 (The parts from 54) = 9; 4+2 (The parts from 42) = 6 "What do you get if you multiply 6 by 9" The answer is all yours.
1932 Is the year when Ford and Arthur return to the present in "Life, the Universe..." and it's also the day that the Vogons came to the earth. Therefore: 1986 (The year of the Copyright) - 54 = 1932!!!
From Nicholas E. Scripture: 50 Mathematical Puzzles and Oddities, Faber and Faber, London,1963, p.47, chapter "Oddments in Algebra", No. 30:
The problem is:
Prove that for any value of 'n', as long as it is a whole
number, (n^7-n) must always be a multiple of 42.
4^2 = 2^4. Other than 24, this is the only non-palindromic two-digit number for which that is true
if you assign the alphabetical value of to the letters of the words: BIG BANG (or GNAB GIB) you get 2+9+7+2+1+14+7 which equals to, indeed, 42
They've found a star that has a planet circling around it for the first time. Distance to this star: 42 light-years
A grad student at UVic says that the lifetime of the universe will be 10^42 years. Note that this is the lifetime, not the age...
Propane boils at -42 C
The maximum temperature in degrees celsius a human can survive in case of a disease is (of course) 42 degrees celsius !
The REAL Hubble is (d / s) distance over speed of galaxies. As we cannot measure the speed accurately, the Constant is not known but scientists think it is betwixt 100 and 40, therefore it could be FORTY-TWO.
If you take the number 'PI' as 3.14, and multiply the whole part (3) by the decimal part (14), 3 X 14 --> 42
The Specific heat capacity of water is 4200 J/kg oC(Joules/Kilograms x degrees celsius)
If you drilled a hole through the Earth from one point to another, and that hole was frictionless, it would take 42 minutes for gravity to move you from you side of the tunnel to the other. That also goes for satellites orbiting the Earth with only gravity moving them.
The galaxy M84 (Messier Catalog) is 42 x 10^6 light years distant. It is an elliptical galaxy in Virgo with a diameter of 25,000 light years
Two physical constants in the universe are the speed of light and thediameter of a proton. It takes light 10 to the minus 42nd power seconds tocross the diameter of a proton.
The Earth is sometimes referred to as the third stone (or rock, if you're a Jimi Hendrix fan) from the sun. A stone (British measurement system) is 14 pounds, so we have 3 x 14 = 42
The world record in memorizing PI is quoted, from the Chicago Sun-Time, to be "over 42,000"
All dogs have a total of 42 teeth over their lifetimes
The average dormancy of an active volcano is 42 years.
The fastest speed for a land animal for 1 mile is 42mph (Guiness, page 59).
Greatest recorded distance covered by one jump of a kangaroo is 42 feet (Guiness, page 71).
The longest lava flow recorded is 42 miles (Guiness, page 129).
The worlds widest conveyor belt is 42 feet wide (Guiness, page 335).
The largest shoes ever sold were size 42 (Guiness, page 361).
A barrel contains 42 gallons.
The time it took for one person to eat an entire ox was 42 hours (Guiness, page 492)
42 occurs infinitely many times in the number PI.
When PI is rounded to the third decimal place contains the number 42
Consider the following: ( ( 1 * 2 * 3 ) - 4 + 5 ) * 6 = 42
In reverse polish notation (postfix), used on most HP calculators (and others), the expression
2 4 2 4 2 * + * +
evaluates to 42! (This, of course, is closely related to the fact that the binary number system, 101010 means 42, or that in base 4, 222 means 42).
SQRT(41*42+SQRT(41*42+SQRT(41*42+SQRT(...))))=42 where SQRT() means the 'square root' of the bracketed expression.
The Efficiency of human respiration is 42
The number of chromosomes of harvested wheat is 42
In the "Tree Information" sketch for Monty Python, the Larch (the one they show over and over again) is tree number 42.
42 in base 4 (222) is the same as in base -5 ie. 54
1 joule is equal to 4.2 calories
The following question appears in the Rhind Papyrus, which is the oldest mathematical book known and originated in Egypt in 1650 B.C.: "If a certain number, two-thirds of it, half of it, and a seventh of it are added together, the result is 97." The number, of course, is 42. (x + 2x/3 + x/2 + x/7 = 97 --> x = 42) This was derived from an 8th grade Addison-Wesley Algebra book, page 444.
42 in base 4 is 222 or 4;222. Also, if you multiply or add the digits together, you get the same thing for 42 and 222.
Posted on: 18 February 2003 by Matthew T
Dobbin,
42 is not the answer (at least not in base 10, MASSIVE hint!), though it may well be THE answer.
Matthew
42 is not the answer (at least not in base 10, MASSIVE hint!), though it may well be THE answer.
Matthew
Posted on: 18 February 2003 by Matthew T
288 has two factors which are only 2 apart...
Posted on: 18 February 2003 by NB
Mathew,
16 & 18
Regards
NB
16 & 18
Regards
NB
Posted on: 18 February 2003 by Matthew T
NB
keep going...
keep going...
Posted on: 18 February 2003 by NB
whats next?
Posted on: 18 February 2003 by steved
Matthew,
I wasn't being funny when I said I was confused - I genuinely don't understand the gist of the problem. Would it be possible to perhaps restate it in a different way, so that the penny drops?
Thanks,
Steve D
I wasn't being funny when I said I was confused - I genuinely don't understand the gist of the problem. Would it be possible to perhaps restate it in a different way, so that the penny drops?
Thanks,
Steve D
Posted on: 18 February 2003 by Paul Ranson
The three Dougs and Richard can make 10 CDPs an hour, the three Dougs and Anna can manage 12. When they all work simultaneously in the time it takes Richard to make one the others make multiples of one. In the time it takes the glue to dry Richard, each Doug and Anna all make whole numbers of players. The total number of players is a square.
Paul
Paul
Posted on: 18 February 2003 by Don Atkinson
Paul,
I can understand your brain teaser.
Like Steved, I am completely unable to understand what Matthew was saying in his brain teaser.
Matthew,
could you amend the wording of your brain teaser to make it unambiguous. Perhaps your brain teaser and Paul's are the same ?
Hope you don't feel offended by my inability to comprehend what you wrote ? Apologies ifyou do.
Cheers
Don
I can understand your brain teaser.
Like Steved, I am completely unable to understand what Matthew was saying in his brain teaser.
Matthew,
could you amend the wording of your brain teaser to make it unambiguous. Perhaps your brain teaser and Paul's are the same ?
Hope you don't feel offended by my inability to comprehend what you wrote ? Apologies ifyou do.
Cheers
Don
Posted on: 19 February 2003 by Matthew T
Paul has summed it up well. You are right, it is a little unclear though maybe not unambigious.
Still waiting to see if anybody can get 288 balls in the box...
Matthew
Still waiting to see if anybody can get 288 balls in the box...
Matthew
Posted on: 19 February 2003 by Don Atkinson
Matthew, (and others),
Still waiting to see if anybody can get 288 balls in the box...
Just the usual re-cap, to coin a phrase !!
Matthew has published, or confirmed, the following facts which I presume we can take at face value, without any ambiguity.
I have a box 5cm x 5cm x 10cm and spherical balls of diameter 1cm. What is the maximium (sic) number of balls can I fit in the box?
I managed 288.
Every wall was touched to get 288 balls in the box and there was no scope for moving them around. (editor's note..ie Don's query. does this mean all six sides, including the top of the box, are touched by at least one ball, in the final solution ? eg one ball in each of two diagonally opposite corners; and that each and every ball touches its neighbours or the sides of the box, such that no ball can move)
288 has two factors which are only 2 apart...16 & 18
There are an equal number of balls in each layer, and each layer is identical, alternate layers line up with one another.
It assumes that the top row of balls cannot exceed the height of the box. Yes (there is a top).
FCC stands for face centred cubic which is one of two ways to get the maximum packing density, the other being hexagonal close packing. The obvious advantage of FCC is that you have a very high degree of symmetry.
Did I miss anthing ???
Cheers
Don
Still waiting to see if anybody can get 288 balls in the box...
Just the usual re-cap, to coin a phrase !!
Matthew has published, or confirmed, the following facts which I presume we can take at face value, without any ambiguity.
I have a box 5cm x 5cm x 10cm and spherical balls of diameter 1cm. What is the maximium (sic) number of balls can I fit in the box?
I managed 288.
Every wall was touched to get 288 balls in the box and there was no scope for moving them around. (editor's note..ie Don's query. does this mean all six sides, including the top of the box, are touched by at least one ball, in the final solution ? eg one ball in each of two diagonally opposite corners; and that each and every ball touches its neighbours or the sides of the box, such that no ball can move)
288 has two factors which are only 2 apart...16 & 18
There are an equal number of balls in each layer, and each layer is identical, alternate layers line up with one another.
It assumes that the top row of balls cannot exceed the height of the box. Yes (there is a top).
FCC stands for face centred cubic which is one of two ways to get the maximum packing density, the other being hexagonal close packing. The obvious advantage of FCC is that you have a very high degree of symmetry.
Did I miss anthing ???
Cheers
Don
Posted on: 19 February 2003 by Don Atkinson
Chaps (as Mick Parry would say !!)
So, Steved and myself thought Matthew's post about glue was unclear and possibly ambiguous.
When I sat my 'O' level maths exam, shortly after Dickens had popped his clogs, and well before Einstein's general theory of relativity has been fully understood by the man on the Clapham omnibus, I had to deal with the following ambiguity.
The railway distance between Newcastle and London is 300 miles. Train A passes through Newcastle at a quarter to ten in the morning and continues towards London at a steady 60 mph. Train B starts from London at 10.4 am and travels towards Newcastle at a steady 75 mph. At what time will the trains pass each other and how far from London will they be. You can ignore the time taken by train B to accelerate to 75 mph.
Unclear and ambiguous questions have been a phobia ever since.
Cheers
Don
So, Steved and myself thought Matthew's post about glue was unclear and possibly ambiguous.
When I sat my 'O' level maths exam, shortly after Dickens had popped his clogs, and well before Einstein's general theory of relativity has been fully understood by the man on the Clapham omnibus, I had to deal with the following ambiguity.
The railway distance between Newcastle and London is 300 miles. Train A passes through Newcastle at a quarter to ten in the morning and continues towards London at a steady 60 mph. Train B starts from London at 10.4 am and travels towards Newcastle at a steady 75 mph. At what time will the trains pass each other and how far from London will they be. You can ignore the time taken by train B to accelerate to 75 mph.
Unclear and ambiguous questions have been a phobia ever since.
Cheers
Don
Posted on: 19 February 2003 by Paul Ranson
I presume the ambiguity is whether 10.4 is 10.40, 10.04 or 10.24?
I didn't think Matthew's question was ambiguous. But the answer I got was rather dull, so I've probably got it wrong.
9 is 4*2+1
Paul
I didn't think Matthew's question was ambiguous. But the answer I got was rather dull, so I've probably got it wrong.
9 is 4*2+1
Paul
Posted on: 20 February 2003 by Matthew T
Some resolution or how fast can you make a CDP!
Richard with the help of the Doug's takes 6 minutes to make one CDP
Anna with the help of the Doug's take 5 minutes to make one CDP
If we take the time it takes Anna to make a CDP as 1/A then A is here productivity etc
1/5 = A + 3D
and
1/6 = R + 3D
which gives
1/30 = A - R
We need Richard to get one area and the others to get a multiple of this so if look at different multiples for the Dougs, i.e. Richard gets one area, each Doug gets 1,2,3... then we can calculate how many areas are required in total and find a solution that gives a square.
so
RX = D (where X is an integer)
then it can be shown that
R = 1/(6(1+3X))
D = X/(6(1+3X))
A = 1/(6(1+3X)) + 1/30
and therefore
A/R = 1 + (1+3X)/5
which gives the total number of areas as
1 + 3X + 1 + (1+3X)/5
which for integer values of X gives integer solutions for 3,8,13,17... and a square number of areas for X=13 of 49 (not unique). Therefore the number of remaining areas is 48, which is 42 in base 12.
Suspect the solution will be a confusing as the question.
cheers
Matthew
Richard with the help of the Doug's takes 6 minutes to make one CDP
Anna with the help of the Doug's take 5 minutes to make one CDP
If we take the time it takes Anna to make a CDP as 1/A then A is here productivity etc
1/5 = A + 3D
and
1/6 = R + 3D
which gives
1/30 = A - R
We need Richard to get one area and the others to get a multiple of this so if look at different multiples for the Dougs, i.e. Richard gets one area, each Doug gets 1,2,3... then we can calculate how many areas are required in total and find a solution that gives a square.
so
RX = D (where X is an integer)
then it can be shown that
R = 1/(6(1+3X))
D = X/(6(1+3X))
A = 1/(6(1+3X)) + 1/30
and therefore
A/R = 1 + (1+3X)/5
which gives the total number of areas as
1 + 3X + 1 + (1+3X)/5
which for integer values of X gives integer solutions for 3,8,13,17... and a square number of areas for X=13 of 49 (not unique). Therefore the number of remaining areas is 48, which is 42 in base 12.
Suspect the solution will be a confusing as the question.
cheers
Matthew
Posted on: 20 February 2003 by Matthew T
Don has summed up the box problem nicely.
I think you should have enough now although it will probably help if you start on the 5x5 base.
Matthew
I think you should have enough now although it will probably help if you start on the 5x5 base.
Matthew
Posted on: 20 February 2003 by Don Atkinson
288 balls - I have a solution (in concept)
I will outline a simple concept. It doesn't provide the right answer but it does provide a useful step forwards.
I will then provide a more complex conceptual solution.
Using the factors 16 and 18 and the idea of using the 5x5 dimension as a base suggests either :-
18 layers of 16 balls each or
16 layers of 18 balls each.
My next two posts will outline each concept. (hopefully)
Cheers
Don
I will outline a simple concept. It doesn't provide the right answer but it does provide a useful step forwards.
I will then provide a more complex conceptual solution.
Using the factors 16 and 18 and the idea of using the 5x5 dimension as a base suggests either :-
18 layers of 16 balls each or
16 layers of 18 balls each.
My next two posts will outline each concept. (hopefully)
Cheers
Don