Brain Teaser No 1

#11 TWO MINUTE TEST (Steved P6)
Answer the following 10 questions IN NO MORE THAN
2 MINUTES - NO CHEATING!!!

1. Do they have 5th November in America?
2. Some months have 30 days, some have 31 days. How many months have 28 days?
3. If you had only one match and entered a dark room where there was an oil lamp, an oil heater and some kindling wood, which would you light first?
4. If the doctor gave you three pills and told you to take one every half hour, how long will they last?
5. A man built a house with four sides, a rectangular structure, each side having a southern exposure. A big bear came toddling by, what colour is the bear?
6. A farmer has 17 sheep - all but 9 died. How many sheep did he have left?
7. Divide 30 by half, add 10. what is your answer?
8. Take two apples from three apples. What do you have?
9. How many animals of each species did Moses take into the ark?
10. If you drive a bus with 17 people on it from London and stop in Birmingham to pick up 9 more and drop off 5 passengers, and at Manchester you drop off 7 more and pick up 6 and arrive in Leeds 5 hours later, what is the name of the driver?

All done. That two minutes went quick!!!



#12 The Boring Shpere - or Another Load of Balls (Don Atkinson P7)
A six-inch long hole is drilled right through a sphere, along a diameter. What is the volume of solid material left in the resultant bead?



#13 Two Numbers (Omer p8)

The question I have is a little more theoretical than the others we had. Still it might interest someone.

Suppose someone chooses two different integer numbers and writes them in an envelope. You pick one of them (by chance) and open it.

Now you have to guess whether this was the smaller or the larger number.
guessing "blindly" will yield 50%. Is there a better strategy (even by a fraction ?)



#14 More Balls (Matthew T p8)

A ball, a cylinder and a ring, each 5 inches in diameter, sit at the top of an inclined plane. If all three objects start rolling down the incline at the same instant, which one will reach the bottom first? (Assume that they all roll efficiently that is, they are perfectly formed and don't wobble--and ignore ant effect of air resistance and friction.)



#15 Seven Pirates (RichardN p9)
There are seven pirates and they have to divide up their stash of fifty gold coins. The pirates have a predefined order of seniority. You are the leader and it is your job to share the gold among the pirates, keeping as much as possible for yourself.

However there is a problem... There isn't much loyalty among pirates. The rule is that if an absolute majority of pirates aren't happy with their share, then they will kill the leader and the new leader will share the gold among the remaining pirates.

What a pirate most wants is to stay alive.
The next most important thing is gold.
Pirates like killing people.
For example, given a choice between getting 10 gold pieces or getting nine gold pieces and killing the leader, the pirate would choose 10 gold pieces.
Given a choice between getting 10 gold pieces and getting 10 gold pieces and killing the leader, the pirate would choose the gold AND killing the leader.

So exactly how much gold can you keep for yourself. You are allowed to explain to the other pirates why your choice is the best result they can hope for...

All the pirates get a vote, including the leader.

#16 Three Card Trick (Don Atkinson P12)
(Note the original text seems to be missing, but replies start at the top of P12)
You have three cards, one is red on both sides, one is black on both sides and one is red on one side and black on the other. They are placed in a bag and moved about until they are random. You put your hand in the bag and pull out one card, taking care that you only see one side of the card. The side you see is red, do you bet one the other side being red or black?



#17 OT: Physics Question (MatthewR P12)
With all this talk of bodies rolling down inclined planes perhaps one of you can explain this for me.

On a cycling trip my friends and I ended up in a conversation about whether weight differences in bikes designed for downhill racing really made any difference since by definition you were going downhill and the riders phyiscal effort was pretty much irrelevant. During this my friend Penny suggested that if we all lined up at the top of a hill, set off together and coasted downhill Dion -- being 17 stone of solid Maori -- would accelerate away.

However, half remembering my schoolboy physics and something about how Galileo dropping cannonballs off the Leaning Tower of Pisa proved that heavy objects don't acccelerate faster we would in fact all accelerate at the same rate and stay roughly level. And if there were any slight differences they would be related to air resistance and so Dion -- with his improbably large cross sectional area -- would presumably, if anything, be slower. So we would all start level until we hit a top speed and then Dion would accelerate away becuase of something to do with the fact that you can't kill mice by throwing them down coal mines.

So we had a bet and then conducted a rigourosly controlled scientific experiment at the next hill. Much to my dismay Dion shot off into the distance. And not just crept away slowly either but shot off in a manner which suggested that some very large differnce was responsible. i.e. most likely his enormous mass. We repeated this a number of times with the same results and even swapped bikes in case Dion's uber expensive Zirconium pedals really did make a difference (They didn't).

So the question is why do fat blokes on bikes seemingly accelerate faster when coasting downhill?



#18 Ladder (BAM P13)
Well, I've been thinking of another puzzle to keep you occupied in the run-up to Christmas. I also wanted one that would foil Don's spreadsheet fetishishm temporarily Here 'tis:
'Twas the night before Christmas and in the Naim factory not a creature was stirring except for... In an attempt to discover the secrets of Naim's new NAP500 a team of Sony engineers (the ones mentioned in that other post who collect the musical electrons off the floor that have fallen out of unsoldered speaker cable strands) decided to gain illegal entry to the Naim factory. Oh the shame of it!
They arrive at the factory in the dead of night and decide to try to gain entry via the roof. Naturally they aren't carrying a ladder because this would arouse suspicion among the locals in Salisbury. So they search for one and eventually find an old wooden, slightly rotten set of rungs. The ladder has length L. The factory roof is hard to access, even with a ladder, and on their first attempt the ladder crackles and threatens to break. They stop climbing and wonder what to do. Scouting for a way in they find a wall against which a large, cubic crate has been left of side 1m. They reason that if they position the ladder so that it touches the ground, the wall and the edge of the crate that this will provide sufficient support to prevent the ladder from breaking. Good.
Being exacting engineers and slaves to planning they first decide to calculate the exact distance that they must place the foot of the ladder away from the side of the crate so that it will touch both the wall and the crate edge. Can they work it out before sunrise?
In terms of the ladder length L what is the horizontal distance the foot of the ladder must be placed away from the side of the crate? Assume the ground is horizontal, the wall vertical and the crate a perfect cube.
The winner will have the most elegant formula.



#19 Orbits (Matthew T)
Imagine...

A planet which happens to have a small donut like tunnel within it which is perfectly centered on the center of the planet, which happens to be perfectly formed. For an object to maintain orbit in the tunnel what is the relationship between it's speed and the distance the tunnel is from the center of the planet.
The tunnel is small enough that the gravitional effect of it on the planet is nil, the planet is not spinning.



#20 A Weighty Problem (Don Atkinson P14)
What is the minimum number of weights needed to weigh up to 40kg in 1kg increments? Using the old 'Scales of Justice' or simple balance type of scales

#21 Another Question (Matthew T P14)
When an airliner is flying at an altitude of 10 km, the temperature of the air outside may be as low as -50ºC. One might think that this would require the use of heaters inside the cabin, but in fact an aircraft flying this high must use air-conditioners. Why?



#22 Fiction, pure fiction (Don Atkinson P15)
How old is Paul S?

Well, God granted him to be a boy for the sixth part of his current life, and adding a twelfth part to this he called him a hansom youth (Fiction, pure fiction); After a further seventh part he took a partner in love and five years later was granted a son. Alas! the wretched child, after attaining the age of half his father's current life, joined Mana as their sales director. Four years have passed since that dreadful day but Paul's consolation has now been realised with the launch of Fraim. To celebrate this glorious event and his xxx birthday, Paul has just decided to deliver a beautiful 5 tier Fraim supporting a nicely run-in NAP500 to Don Atkinson......Ok, Ok, I SAID it was fiction, pure fiction......

So, according to the facts above, how old is Paul?? (Lets hope his sense of humour is just as great!) Fiction, pure fiction so its no good ringing Naim, even Paul doesn't know how old he is - yet!



#23 An intermission (Omer P15)
Two paratroopers land on an infinite one-dimensional road. They leave the parachute where they land, and have to meet the other.
To this end (meeting) they both have to use exactly the same directions.
These directions include ONLY the following "buildstones":
1. go north one meter.
2. go south one meter.
3. goto direction step # (e.g., goto step #2)
4. if (met parachute) goto (direction step #)
How can they meet ?



#24 Algebraic Multiplication (BAM P16)
What does this product equal?
(x-a)(x-b)(x-c)...(x-y)(x-z)



#25 A Truel (Don Atkinson P17)
What's a truel? I hear you ask. Well, it's like a dual, but there are three adversaries rather than just two !

Imagine Stallion and Vuk throwing the old gauntlet on the ground, followed by Mick Parry stepping in to try and calm things down, but somehow getting involved! We'd have a three-way shootout on our hands. Old Martin Payne could act as referee. Mind you, Martin would have to take care where he stood. Remember also, both Stallion and Mick have gun collections, but I'm not sure about Vuk - bit of a disadvantage using a camera for this type of shoot, even if it is a Leica. Come to think of it, wasn't Mick thinking about a Leica?

Anyway, back to the truel. Rather than a free for all, and to give Martin a sporting chance as referee, the rules are that each 'player?' takes one shot at a time, with a free choice of target. Clearly survival is the key to winning and based on past performance, we know that Vuk hits his target only 1 in 3 times. Mick Parry hits his 2 times out of 3. Stallion of course always hits his target, never misses. (Hey this is only a story! It might seem like a dream to Stallion and a nightmare to everybody else, but it's still only a story!)

Now to give Vuk a chance, Martin decides to give him first shot. What is Vuk's best strategy for survival?

Don,

So will the Unofficial Naim Teaser Book be out for the Christmas rush?

- Dan

So will the Unofficial Naim Teaser Book be out for the Christmas rush?

Ha Ha.....if only.

But, Dickens published his books as installments, and the next 25 teasers will be on their way soon.

Watch this space

Cheers

Don

#26 An easy mistake !! (Don Atkinson P18)
I spent an hour this evening looking for an insight to Bam's ladder problem. Found myself making some silly mistakes. A bit like the following one, where it appears possible to prove that 1 = 2.

Let's start with the simple statement a = b
Multiply both sides by a
a^2 = ab
Add a^2 - 2ab to both sides
a^2 + a^2 - 2ab = ab + a^2 - 2ab
Simplify
2(a^2 - ab) = a^2 - ab
Finally divide both sides by a^2 - ab
2 = 1

Somewhere there is a disastrous error!



#27 It all adds up (BAM P18)
I was very amused many years ago when my A level maths teacher drew a relationship among some of the most hideous numbers used in mathematics. It was like combining a pile of abstract, intangible mind-bending constants and miraculously getting something pure and tangible.

Basically, he showed an equation that related e, pi, i and 1. Do you know what it is?



#28 And another age-old problem (Don Atkinson P18)
But easy, just to while the time away, waiting for ladder inspiration!
The ages of my father my son and myself total 85 years. My father is just twice my age, and the units figure in his age is equal to the age of my son.

How old am I?



#29 A Pair of 135s for Xmas (Don Atkinson P20)
The Sony engineers have used the ladder and now got over the wall (they used my 'neat' formula !! not Bam's elegant one) and are now in the warehouse.

Now remember its Xmas, so the warehouse is virtually empty 'cos most of the kit was all delivered to Grahams and other fine dealers in time to fill our Xmas stockings. Anyhow, there are eight 250s and two 135s scattered around the room (randomly). But the room is pitch black and the boxes are identical as are the weights of each package. So it’s a genuine game of chance. The Sony guys can only escape with TWO boxes. They really want a pair of 135s.

What's their chances of picking them?



# 30 Sackfuls of Apples (Don Atkinson P20)
Stallion, Vuk and Mick Parry are pinching apples in Blzebub's orchard. (well Mick's diplomacy has been successful in getting everybody to be friends except Blzebub). Stallion picks 7 sackfuls containing 16kg each, Vuk picks 7 sackfuls each containing 14kg, Mick has smaller sacks and he picks 10 sackfuls containing 9kg each. They had agreed beforehand to share the fruit equally.

How can they do this without opening any of the sacks?



#31 Try Counting Sheepdogs!! (Don Atkinson P20)
Bill, the shepherd and Shep, his dog, are going home after a day on the hills. Bill walks at a steady 4 mph. When they are half a mile from his cottage, Bill sends Shep on ahead to warn his wife he is on his way. Shep races to the cottage, barks, immediately returns to his master and continues to run back and forth between the cottage and Bill until Bill reaches home. Shep's running speed is 16 mph.

How far did he run altogether, from when he was sent on ahead?
Perhaps if you try this one 'in your head', you'll fall asleep in no time at all.



#32 Hi-Fi Costs £1,000 (Don Atkinson P20)
A hi-fi dealer sells Naim hi-fi which is priced at, CD player £160; amplifier £230 and speakers £390; and Sony hi-fi where the prices are CD player £170; amplifier £240 and speakers £400. A customer buys some of this hi-fi and it costs him exactly £1,000.

What does he buy?

Hint, like in real life, there is absolutely no rationale about what he buys!!



#33 An Unlikely Story (Don Atkinson P20)
At the Bristol Hi-fi show a group of six Forum members won the chance to visit the Naim factory, to select the product of their choice, but there was room for only four of them, and they had to reach agreement who would go. (Look, I said it was an unlikely story!)

Marcus wouldn't take part unless Steven was allowed to take part
Steven wouldn't be in the party if Vik was.
Blzebub wouldn't be in the party if both Marcus and Steven were in it.
Vik wouldn't be in the party if Mick was allowed in
Mick will join the party with any of the others
Joe won't be in the party if Marcus is, unless Vik is in it too.

Which four members visited the factory? and what was each one's item of choice?

BTW, any similarity between these fictitious characters and members of this Forum bearing similar but equally fictitious names are entirely in-coincidental.



#34 Another Brain-strainer (BAM P21)
You have been selected as the speaker cable supplier to Naim Audio (a lesser known but highly reputed hifi company in an ancient druid village with an unusually well-endowed church in Wiltshire). Your company manufactures high purity, extruded copper wire that has been processed to achieve near single-crystaline structure.
Naim requires a speaker cable that consists of two parallel multi-strand conductors, separated by a fixed distance and encased in flexible black plastic. Furthermore, Naim requires that you print the Naim logo on the sleeve and chevrons to indicate the correct orientation of the cable when in use. Being a good salesman you simply agree to all of this whilst remaining po-faced.

You have a meeting with your production supervisor and start to flesh out the detailed specification. How many strands are needed per conductor? What should the separation be between the conductors? How thick should the PVC sleeving be? All questions are quickly resolved and all is going well until the supervisor asks you: "how do we decide which direction to put the chevrons in?".

You hadn't thought about this. Your gut instinct is that it doesn't matter as long as there are chevrons and that is what the client requires. However, your supervisor is an engineer and a bit of stickler for process and far too anal to understand the compromises of the wider business environment. So you are going to have to fob him off with some technical rather than marketing reasoning.

The supervisor senses your hesitation and glares at you. He says the only possible directional aspect is the direction that the wire has been drawn through the extruder. He bluntly asks you to define, relative to the chevron direction, which wire orientation you want for each of the strands within each conductor, the direction of twist of the strands within each conductor (clockwise or anti-clockwise when viewed in the direction of the chevrons) and the relative orientation of the two conductors.

What answer should you give the supervisor in order to achieve the best sounding cable for Naim Audio?



#35 Primes (Omer P22)
Prove that every even number greater than 2 is a sum of two prime numbers.
I have a solution, but it's too long to write here, and it's actually for a different problem.



#36 A simpler problem... straight from my daughter's maths book... (Ken c P22)
in the attached diagram, the large circle has diameter 10cm. each of the inner circles are identical size and they each touch the large circle and 2 adjacent circles symmetrically as shown (not very well i a'm afraid). non of the circles "cross" each other -- diagram not very good in this respect

find the radius of the largest circle which will fit in the middle.

enjoy

(PS, You will need to look at P22 to see Ken's diagram)



#37 And another little, simple one... (Ken c P22)
prove that the product of 4 consecutive numbers is always one less than a square number.

so for example: 2 * 3 * 4 * 5 = 120 which is 1 less than 11^2
enjoy



#38 Another one just to keep you away from the ladder problem... (Ken c P23)
prove that the product of any 3 consecutive integers is a multiple of 6.



#39 Five circles (Don Atkinson P23)
Now, i haven't thought the next variant through at all (believe me!) but if we put five (5) circles inside the big one, do you recon there's a "golden ratio" in there somewhere?



#40 The magic of numbers... (Ken c P23)
this is NOT a puzzle. its just something i read about which is rather interesting, but i am not sure i can "explain" it. number theory has a lot of very interesting/intriguing results.
well, here goes:
take the number 142857 and multiples of it as follows:
142857x1 = (to start you off...)
142857x2 =
142857x3 =
142857x4 =
142857x5 =
142857x6 =
what do you notice??
now arrange the digits of the results of the multiplications (1 row per result) as a 6x6 matrix and compute the row sums and column sums. curious isn't it??
any ideas??
in fact you can continue the pattern beyond 6 and there will be interesting things to observe?



#41 What's gone wrong??? (Ken c P23)
let x = 0.99999...
then 10x = 9.9999...
i.e 10x - x =9.9999... - 0.9999...
i.e 9x = 9
i.e. x = 1
what !!!???



#42 This gets stranger still. (Matthew T P24)
try 1/17!
and 2/17....
freaky



#43 Another little puzzle, I don't know the answer!? (Ken c P25)
heard in the pub with a few customers -- was well gone by then (my excuse and i am sticking to it!!)

a lorry is travelling in a straight line at constant speed. it starts turning so that the front wheels describe a circle of some radius.
question: is the imaginary line along which the lorry was travelling originally a tangent to this circle?

if so, why so, if not, why not...



#44 Just to keep you away from paraboli and hyperboli... (Ken c P26)
do the following multiplications and tell me what you notice. interesting isnt it?? another one of those fascinating things you can do with numbers -- i havent thought about it too deeply, but i doubt i can explain WHY it happens. well here goes:
2 + 9 * 1 =
3 + 9 * 12 =
4 + 9 * 123 =
5 + 9 * 1234 =
etc. obviously do the multiplication before the addition ("bidmas" or something...)



# 45 Golden ratios (Don Atkinson P26)
Based on ken c's idea draw:-
1 large outer circle, inside which we draw
5 middle circles, each of which touch the outer circle and the two ajacent middle circles
1 inner circle which touches each of the 5 middle circles
overall picture looks a bit like a wheel bearing with 5 roller bearings!
We have already learned how to calculate the diameters of the middle and inner circles given the diameter of the larger circle (and the number of roller bearings)

Try calculating :-
(a) distance (centre to centre) between two ajacent middle circles
(b) distance between two non-ajacent middle circles

Compare ratio of a/b and b/a and post comments??



#46 Tying Nanny up (BAM P27)
I fear Ken may have a breakthrough brewing for the ladder puzzle so here's another problem as an interlude.

Near a small town with a big church in Wiltshire, an audio industry worker named Paul supplements his meagre salary with part-time farming. Paul owns a circular field of grass of unit radius and he also owns a very hungry goat from which he derives milk for cheese. Paul wishes to tie the goat to an existing post on the circumference of the field and wants to manage the growth of the grass so the goat doesn't decimate it too quickly. He therefore decides to use just enough rope so that the goat can only eat 50% of the grass.

What must the length of the rope be? Assume the distance between the jaws of the goat and the point of attachment of the rope is insignificant.



#47 (Probably BAM P27)
evaluate(x-a)(x-b)(x-c).......(x-z).



#48 My next one falls into the dead easy class. (Don Atkinson P27)
a) which is smaller 973x973 or 972x974 ?
b) elaborate !



#49 Who should you marry? (BAM P28)
It's your lucky day - maybe!

A great king, who's hifi system you have improved by planting a solid gold earth rod in the moat of his castle, has decided to reward you by giving you a chance to marry one of his 10 daughters (personally I blame the estrogen in the rivers).

You will be presented with the daughters one at a time and, as each daughter is presented, you will be told the daughter's dowry (which is fixed in advance). Upon being presented with a daughter, you must immediately decide whether to accept or reject her (you are not allowed to return to a previously rejected daughter).

But there is a catch. You didn't think it would be this easy to marry a king's daughter did you? The king will only allow the marriage to take place if you pick the daughter with the highest dowry.

What is your best strategy and what is your chance of getting hitched?



#50 Another set of circles (Don Atkinson P30)
Its pretty obvious that the maximum number of different sized circles (excluding infinitely large circles that have become straight lines!) that can be made to 'just touch' one another is four. (each and every circle just touches the other three).

In the attached diagram, the three circles A, B and C have radii of 1, 2 and 3 respectively. What are the radii of the two 'fourth' circles, that can be made to just touch each of A, B and C. (note the two 'fourth' circles will not touch each other, cos that would break the golden rule!!)

Oh! BTW, there is a standard way and an elegant way to solve this little problem. The normal way will do for most, but in the case of Bam.....

#28 And another age-old problem (Don Atkinson P18)
But easy, just to while the time away, waiting for ladder inspiration!
The ages of my father my son and myself total 85 years. My father is just twice my age, and the units figure in his age is equal to the age of my son.

How old am I?

You are 26, your father is 52 and your son is 7

Don,

Indeed a momentous occasion and I look forward to whileing away the hours trying to remember how to solve the teasers that I myself have set, let alone others! However to conclude on the Mick P bicycle hill...

1 mile is 1609.344m
So each rotation of the pedal takes him 4.73m (1609.344/340)
The length of the crank is 170mm so each rotation of the pedal is 1.068m

So, when Mick has maximum leveage, full body weight on pedal, will the bike role up the hill or down.

The maximum slop up which he could get the bike to move upwards is the one at which his vertical movement upwards is less then his bodies movement downwards (relative to the bike) and can be expreesed as arcsin(1.068/4.73) which is 13 degrees. So there is no-way Mick could get the bike up the hill (unless he had cleats on, even then it would be the story of greek legends!), in fact it would be neigh on impossible to get the bike up anything above 10 degrees, as the optimal loading condition is only true for a small proportion of the pedal stroke.

Look forwrd to the rest of the brain teasers.

Matthew

Su Doku

I spotted a new puzzle in the Times recently. It's explained neatly at http://www.sudoku.com

Solving is rather a trivial mechanical process, I don't see it being interesting to do on a regular basis.

Anyway I was wondering how many puzzle grids there might be. And if one defined a set or group of grids as those related by reflection, rotation or swapping rows or columns (obviously we can only swap 1,2,3 or 4,5,6 or 7,8,9) how big such a set might be and how many sets of solutions there are.

I've no idea of the answers, but I thought it was more interesting to think about than solving the puzzle...

Paul

On a parallel thread it has been asked whether 1 is a prime or not.

I have posted...

Whether 1 is a prime or not is almost irrelevant and virtually semantic.

"a number that is only divisible by itself and 1"

Our invention of words excludes the number 1, otherwise we would have tortology

Of course we could extend the definition with either "including 1 itself" or "excluding 1 itself".

However, n! + 1 = a prime
(but doesn't generate all the primes and doesn't "prove" whether 1 is a prime or not)

Now, allow me to invite you to the "brain teaser" thread to see whether anyone can explain why n! + 1 = prime (and BTW, this would then demonstrate that there are an infinite of primes)....

Anybody want to post their thoughts ?

Cheers

Don

Paul,

I had a quick look at the link to sudoku and agree that figuring out how many arrangements are possible appears to be a challenge.

Hopefully, vast numbers of new mathematicians will be tempted to browse this thread following my post from the "prime" thread, and we will get a few answers........

Cheers

Don

It looks like a prime cock-up

I will have to check my source, but it looks like I have made a bit of a cock up on the prime number count....

Thanks Omer, for keeping us all on the staright and narrow...

Cheers

Don

found this all the way down on page 5. Anyway I saw this teaser on the way home today on the subway in Toronto.

If you write a 5 letter word (train) in a diamond:

        N
      N I N
    N I A I N
  N I A R A I N
N I A R T R A I N
  N I A R A I N
    N I A I N
      N I N
        N

How many times can you write out train by joining adjacent letters together, two examples given below:

    N I      
      A R      
        T R A I N

And part two: is there, and if there is what is, a generic formula for the number of times a word can be written out given a word of length n is written out in this pattern?

I guess 60.

Matthew

A simple Christmas one

The naim Christmas tree is up repleat with minature CDS3's and 300 etc hanging tastefully from the branchs. However the number of SNAIC's used to connect the hicap to Naim logo fairy lights and to the main logo at the top are not enough (last years tree was half the height) and given that it is wrapped around the tree in a helix the lads in Salisbury are trying to figure how many more they need, assuming that the tasteful (ahem!) fairy lights are not spread further out. Can we help them out, they used ten last year (remeber, one for each fairy light).

Matthew

Preferential treatment.

Thanks Adam, I feel very privileged.

It also feels like coming home (say) from a 5 year trip to the South Pole.....see the first post 101 pages back..........

Cheers

Don