Brain Teaser No 1

Tried and failed

But will try again

Cheers

Don

Job assessment test for printed circuit board designers at Naim.

Square ABCD has a point P within its perimeter.

AP = 10
BP = 2
DP = 14

How long is CP

Note ABCD are labelled clockwise around the perimeter of the square

Cheers

Don

Don,

I think CP^2 = 14^2 + 2^2 - 10^2 = 100

so, CP = 10

cheers,

Dan

How about this?

         Matthew                   Mick
Time   Speed   S(avg)   Dist     Speed   S(avg)   Dist
   0       1       0       2
   1       3       2       4
   2       5       6       6
   3       7      12       8
   4       9      20      16       0        10    16.5
   5      11      30      17      16.5      12    17.5
   6      13      42      18      34        14    18.5
   7      15      56      19      52.5      16    19.5
   8      17      72      20      72        18    20.5
   9      19      90      21      92.5      20    21.5
  10      21     110      22     114        22    22.5
  11      23     132      23     136.5      24    23.5
  12      25     156      24     160        26    24.5
  13      27     182      25     184.5      28    25.5
  14      28     210      26     210

I haven't read the problem, so I've no idea at all!

Paul

Don,

An image of how my 'reply' window looked during the composition of that last message is here.

Paul

quote:

---

Originally posted by Don Atkinson:
_Clean-up time for a couple of outstanding teasers....._

_Number flushes_

_Can you write the numbers 1 to 9 in such a sequence such that it isn't possible to extract from the sequence ANY four-digit rising or falling subsequence?_

_eg 1 2 3 5 4 6 9 7 8 ISN'T acceptable because 1 2 3 5 is a rising subsequence as is 1 2 3 4 and 1 2 3 9 etc etc. But there are no four-digit falling sub-sequences in this particular sequence._



Two-Sheds said

how about this then...

3 2 7 5 1 4 9 8 6

Which is a good answer.



I suggested a (slightly) more elegant solution existed

Well, in fact there are several eg

3 2 1 6 5 4 9 8 7

_ie three, triple reverse runs_

Cheers

Don

---

Of course the next step is find a similiar sequence of numbers that has no 3 digit falling or rising sequences.

Is it possible?

Hopefully this will look better!!1

Looking back at the Matthew and Mick Bike teaser, I realised that I haven't posted my "Trial & Error" solution. So here goes

I created a tabular format to keep tabs on the numbers and I calculated the speed, average speed and distance for each cyclist at 1 second intervals. Dimensions are secs, m/s and m. The green light is at (time 0) and (dist 0) for Matthew and at (time 4) and (dist 0) for Mick.

As if by magic, most of the numbers are whole digits and the time at which Mick passes Matthew and the time at which Matthew passes Mick are nicely on whole seconds.

 
T		Matthew				Mick
Time		Speed	S(avg)	Dist		Speed	S(avg)	Dist
0		1		0

1		3	2	2

2		5	4	6

3		7	6	12

4		9	8	20		16		0

5		11	10	30		17	16.5	16.5

6		13	12	42		18	17.5	34

7		15	14	56		19	18.5	52.5

8		17	16	72		20	19.5	72

9		19	18	90		21	20.5	92.5

10		21	20	110		22	21.5	114

11		23	22	132		23	22.5	136.5

12		25	24	156		24	23.5	160

13		27	26	182		25	24.5	184.5

14		29	28	210		26	25.5	210

Of course, there is no guarantee that this table will copy neatly into the forum. Hope.

The table shows that Mick passes Matthew 72m from the light. This happens 8 secs after Matthew has passed the light. At this time Matthew is doing 17m/s and Mick 20m/s

The "dist" is got simply by adding up all the preceding average speeds. Eg after 8 secs Matthew has travelled (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16) = 72. In practice its even easier because all you do is add 16 to the previous distance of 56....
The next part of the table shows Matthew passing Mick 210m from the light. This happens 14 secs after Matthew has passed the light. At this time Matthew is doing 29m/s and Mick is doing 26m/s.

If it does look better, Thanks to Paul.

If not......well, back to the drawing board

Cheers

Don

Thanks Paul,

Cheers

Don

Dan,

You are spot on with CP = 10

Also with CP^2 = BP^2 + DP^2 - AP^2

Do you ( or anybody else) want to share with others how to "prove"
CP^2 = BP^2 + DP^2 - AP^2
(Assuming that we accept Pythagoras)

Cheers

Don

Matthew,

Of course the next step is find a similiar sequence of numbers that has no 3 digit falling or rising sequences.

Is it possible?

Are you asking, or telling?

First thoughts are that its not possible......

Cheers

Don

quote:

---

Originally posted by Don Atkinson:
Do you ( or anybody else) want to share with others how to "prove"
CP^2 = BP^2 + DP^2 - AP^2

---

Don,

A little tricky without pencil and paper, but if one draws perpendiculars from each side that pass through "P" then I think it's clear how the proof goes.

Dan

Dan

A little tricky without pencil and paper, but if one draws perpendiculars from each side that pass through "P" then I think it's clear how the proof goes.

I know exactly what you mean Dan when you say tricky, please bring back the paper clip facility....and well done with the explanation, spot on.

For the record, this was my explanation.

Assume AB is the horizontal side at the top
draw perpendiculars from each side that pass through "P"...
label the mid points of AB and DA as m and n respectively

 
Am = w;  mB = x
Dn = y;  nA = z

AP^2 = w^2 + z^2
CP^2 = x^2 + y^2
AP^2 + CP^2 = w^2 + x^2 + y^2 + z^2

BP^2 = x^2 + z^2
DP^2 = w^2 + y^2
BP^2 + DP^2 = w^2 + x^2 + y^2 + z^2

AP^2 + CP^2 = BP^2 + DP^2
CP^2 = BP^2 + DP^2 - AP^2
     = 2^2 + 14^2 - 10^2
     = 4 + 196 - 100
     = 200 - 100
     = 100
CP   = 10

Cheers

Don

quote:

---

Originally posted by Don Atkinson:
Matthew,

_Of course the next step is find a similiar sequence of numbers that has no 3 digit falling or rising sequences._

_Is it possible?_

Are you asking, or telling?

First thoughts are that its not possible......

Cheers

Don

---

Well, I think it is possible. Actually I have a set of numbers which seems to fit the bill, so can you find the sequence?

Matthew

Matthew,

I have to admit that I am at a complete loss as to finding an arrangement of the digits 1 - 9 such that there aren't 3 rising or falling digits to be found.

The more I have tried, the more I have found it impossible. In fact, I have a feeling that with more than 4 digits, it is impossible ie 1 to 5 can't be arranged to avoid at least one sequence of three rising or falling digits.

eg 2 1 5 3 4 fails on several counts eg 2..3 4 or 1..3 4
and 3 1 2 5 4 fails on at least 1 2 5 or 1 2..4

Anybody got any ideas....help !!

Cheers

Don

OK, I see the problem. I was looking for continuous sequences, not interrupted ones! That being the case you are right.

So I'm sure you can find a sequence that has no continuous sequences in it.

I guess I must apologies if I have caused any headaches!

Matthew

Mick P has finally decided on a new bicycle and has happily been cycling round Swindon enjoying the autumnal weather. However, in an attempt to obtain that Greek God look he has decided to head out to the Cotswolds for a more strenous ride. He comes across a road sign that indicates 14 degree up hill slope coming up.

Mick had decided on the fixed wheel option and out of curisoty he had noticed that it took 340 turns of the pedals to complete the one mile between two of the more interesting roundabouts in Swindon. He also knew that he had a standard gear set, and the cranks where the 170mm versions.

He was concerned he wasn't going to be able to make it up, even thopugh he was feeling in really great shape! Were his fears founded?

Matthew

rising sequences - I thought about this for a while and could not think of any sequences of 9 numbers with no rising/falling sequences of 3 or more.

I then cheated and wrote a little program to do a brute force search of all possible sequences and didn't find any so I would agree that it isn't possible although there are quite a few that meet the original problem.

Matthew,

He was concerned he wasn't going to be able to make it up, even thopugh he was feeling in really great shape! Were his fears founded?

I'm sort of stuck.....

a bit of a push might help

Cheers

Don

quote:

---

He was concerned he wasn't going to be able to make it up, even thopugh he was feeling in really great shape! Were his fears founded?

---

I worked out a rough gearing of 39/16 which means that he's going to struggle on a 14deg incline.
(based on 4.7m per revolution and 2m wheel circumference, crank length is irrelevant).

John,

I worked out a rough gearing of 39/16 which means that he's going to struggle on a 14deg incline.

I'm no biker, so the next few lines are pure waffle....

I had estimated a rough gearing of 4.4 based on the number of turns of the pedals and the distance travelled.

Not being a biker, I have no idea whether 39/16 or 4.4 would make a 14 deg hill a "toughie"

I was also sure that the wheel diameter AND the ratio of the big sproket (pedals) to the little sproket (wheel) could affect the outcome.

....not mention Mick's physical prowes....

Remember, I have met Mick a few times.....

But if YOU (being a bike man) say he might stuggle, I won't disagree.

Cheers

Don

quote:

---

I had estimated a rough gearing of 4.4 based on the number of turns of the pedals and the distance travelled.

---

1 mile = 1600m 1600/340 = 4.7 unless you're a Roman then 1 mile = 1479m 1479/340 = 4.4

So each turn of the pedals makes you go 4.4m.
I've also cheated by saying the circumference of the wheel is 2m when it's closer to 2.1. So Mick's gearing is slightly lower than what I've claimed.

quote:

---

But if YOU (being a bike man) say he might stuggle, I won't disagree.

---

For way of comparison the pros would probably ride a similar incline using a 21 or 23 rear cog.

Now I think I'll go back to lurking...

[This message was edited by John Sheridan on Sat 13 November 2004 at 23:12.]

[This message was edited by John Sheridan on Sat 13 November 2004 at 23:13.]

John,

Now I think I'll go back to lurking...

...what a shame, just as I was begining to realise there might be more to bikes than meets the eye.....bit like my old "Sturmey Archer ?" 3-speed internal gear hub....

Cheers

Don

Three Years.[/

The Brain Teaser has now run for three years, 2000 posts and 100 pages. It has attracted over 25 contributors, many of whom have set their own questions. I have my own favourite questions including BAM's Ladder, Matthew T's 10x5x5 box and the Monty Hall Goat, (which infuriates CEOs and decision-makers) I'm sure other contributors have their favourites.

However, I realised that newcomers to the forum might not relish the search of 2000 posts to find teasers to their liking. What if I trawled the topic and published a complete (E&OE) list of all the questions to date, in a single, easily accessible block ? This would allow anybody to browse the list and tackle those teasers that appeal.

I have now done this. There are over 200 teasers. I have compiled them into blocks of 5., in chronological order, in 40 odd posts that follow. I have quoted the page in which the original teaser appears so that hints and answers can be found if needed.

I hope this encourages more members to tackle some of the teasers to date, even if they don't want to post their trials and tribulations....

As ken c would say

"enjoy"

Cheers

Don

#1 The Explorer (Don Atkinson P1) 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?



#2 Small Reservoir (BAM P1)
A woman is sitting in a boat in a small reservoir. In the boat with her is a hemp sack containing the dead body of her audiophile husband. She had become frustrated with her obsessive partner mainly due to the proliferation of ugly black boxes and metalwork that had taken over her living room. Not to mention the brick dust and ill fitting carpets where he installed the make-shift mains spur; nor to mention the devastation of her beautiful flower beds where he had insisted on driving a dozen copper pipes into the ground. The final straw came when, at the recommendation of an old school friend in the business (a friend before she met HIM), Ideal Home dropped in to offer to do a feature. Shocked by the hifi carnage they declined to do the feature.

Anyhow, I digress. She has weighed the sack down with two 135s and the twisted remains of his Mana stands and sewn it tightly with NACA5 (ensuring correct direction in the weave) to ensure he will sink without trace. Whilst securing the parcel she glaces over to the reservoir wall and notices a water level marker. With a major struggle she manages to heave the sack of hubby and hifi over board. After a while she settles back in the dingy and smokes a well-earned cigarette. The reservoir wall catches her eye again; she notices the height of the water line has changed.

Has the water risen or fallen against its original level on the water level marker?




#3 Game Show (BAM P2)
Alright, here's an old one so I hope you haven't heard the answer already. If you haven't it is quite interesting. Enjoy!

Late one cold, rainy night in November after a few pints and a rather dubious curry you flop down onto your favourite listening couch and turn on your Naim system. Ah...that's better. But even the PRaT isn't enough to keep your eyes from closing and you soon fall into a deep, deep slumber and start to dream..
.
Congratulations! You have been selected as a participant in a very popular national TV gameshow. This is your first time on TV and on such a show and you are overwhelmed with excitement and nervousness. The gameshow host is none other than Paul Stephenson. Dressed in a gold lame suit with sequens, standing centre-stage on a Fraim. The grand prize: a Naim NAP500 and installaton in your home by Roy George himself. Your heart pounds with excitement.

You find the first rounds easy and before long the other contestants have been eliminated and you have made it to the final challenge. The audience go wild! You try hard to look at them through the bright spot lights and recognize them as your chums from the Naim Forum - all clapping and cheering you on. What a night!

Now the final challenge begins. The lights dim and the cheers of the crowd die down. Paul shows you three doors numbered 1, 2 and 3. He says that the NAP500 is hidden behind one of the doors. You must choose which one. If you choose correctly you win the NAP500. If not you win Paul's hideous suit. The tension mounts. The audience can't contain themselves: shouts begin..."door 1"..."no door 2"...etc. You hesitate. Paul urges you to make a choice. Finally you choose door number 3 (after your existing Nait 3 system). A green light comes on over door 3. Oh no...did I choose wisely...
A hush comes over the theatre as Paul walks over to door number 1 and motions to open it. The atmosphere can be cut with a knife. Slowly, Paul opens door 1 to reveal nothing! Phew that was lucky. The crowd bursts into a relieved cheer. But soon Paul has gripped the handle of door 2 and the crowd go silent. You can hear a pin drop.

The tension is unbearable...

Suddenly Paul stops turning the handle of door 2 and turns towards you. He says he is in a good mood tonight and wants to help you as much as he can. He offers to do you a favour and allow you to change your selection. You can now stick with your original choice...door 3 or switch to door 2. It is up to you and you have just 10 seconds to decide. The audience erupts! "door 2 go for door 2"..."no, no stick with door 3"...

You have 5 seconds left. You are sweating. Everyone is hanging on your next words. Do you stick with door 3 or switch to door 2?



#4 Back to Boats (Dave Cattlin P2)
Three men in a boat with four cigarettes and no matches - how do they get to smoke?
Holy brainteaser



#5 Black or White Hats (Omer P4)
Paul Stephenson, Paul Desmond and Paul Darwin sit together, and out of a very large bag, a black or white (50%) hat is put over each of their heads.
Each of the 3 can see the colours of the other hats, but not the one over his own. They are not allowed to talk, wink etc., but they can coordinate in advance, before the hats were put.
Then each of them gusses the colour of his own hat by writing it in a note, and these notes are opened later after all gusses were made. A guess can also be a pass, "I don't know".
Now, a new NAC122 will to awarded to each if :
1. At least one of them gusses correctly the colour of his hat, and
2. None of them guesses wrong.
What is their best strategy ? Obviously 50% success is possible : they agree that only Paul S. gusses "black", the others pass.
The question is - do they have a better strategy than that ?

#6 The Baron's Treasure (Don Atkinson P4) Baron Von Stephenson kept his gold treasure (fairly got by manufacturing hi quality electronics) in a treasure house. In each room of the treasure house were as many chests as there were rooms. In each chest there were as many gold coins, as there were chests in that room. All the gold coins were of the same size and value.

When he died (ok, this is a futuristic story set in the year 2222), the Baron's will was that his favourite Forum member (Guesses on a postcard please!!) should receive one chest of gold coins. The remaining coins were to be divided equally between the Baron's three forum administrators, Paul Des, Doug G and Paul Dar .
The three administrators were proud and fierce men who would definitely resort to bloodshed if the coins could not be divided equally.

The question is simply: -
a) Was there blood shed.
b) Was there no blood shed.
c) Is there no way of knowing whether there was blood shed or not.

PS. The proof is the real requirement, not simply a one in three guess.



#7 Odd Ball? (Don Atkinson P5)
I have twelve balls that LOOK identical but one (and only one) is either heavier or lighter than the rest. Using a pair of scales, I can compare the weight of any combination of balls that I chose. What is the MINIMUM number of weighings that I need to make, in order to identify the odd ball and decide whether it is heavy or light?
PS 'scales' = those old fashioned ones that look a bit like the Scales of Justice



#8 How Far is it to Salisbury? (Don Atkinson P5)

Ok guys,

Whilst you are trying to figure out how to do it in three (and I think it's very decent of Duncan F and others to sit tight whilst yet others try to figure it out) here's a much more straightforward one.

Naim have two delivery vans that shuttle between Salisbury and their favourite dealer. (Ok, ok, in the original version it was 'two ships operate a ferry service across a river….')

Naim have two delivery vans that shuttle between Salisbury and their favourite dealer. On a particular day, one van leaves Salisbury bound for the dealer, whilst at the same time the other van leaves the dealer, bound for Salisbury. Both vans follow the same route and pass each other (giving the famous 'Naim Salute') 30 miles from the dealer's. At their initial destinations each van takes 15 minutes to load/unload. On the return journey, following the same route, the vans pass each other (giving the famous 'Naim Salute') 15 miles from Salisbury.
How far is it from Salisbury to the dealer's ?

Assume that each van travels at a constant speed and that acceleration/deceleration is allowed for in the 15 minute turn round time.



#9 Another hat related one : (Omer P5)

8 people stand in a queue, all facing one direction like this.
-> -> -> -> -> -> -> ->
Hats (b/w) are layed on their heads. Person #1 sees all other 7, #2 sees the 6 in front, and #8 sees noone.

Starting from #1, each guesses aloud the colour of his hat.

The question is - Do they have a strategy that ensures them some minimum of right guesses ?



#10 Deliberately left blank