Being a big fan of duct Jason has twelve brand new rolls out in the garage, but unfortunately none of them say how much tape is on a roll! To help Jason plan out a big duct tape project can you figure out how much tape is contained on each roll? You are able to measure that the tape is wrapped around a cardboard tube with an outer diameter of two inches, and that the tape wraps around the tube until it has an outer diameter of 3.5 inches. The tape measures out to be 0.01 inches thick. What length of tape is contained on each roll?
The wording needs a bit of work – specifically I want to add some reason why Jason wants to know how much tape is on each roll – but for now I am stumped. I originally wrote it with him wanting to make a suit out of duct tape for his high school reunion – but the absurdity eclipsed the problem. Eclipsing the problem is sort of an ongoing problem when I write this stuff up.
My main problem here, I think, is that it is a very simple problem if you have the tools to solve it – but I am not sure if it is too difficult if people have never worked with power series. I wrote up the solution as an introduction to them – and that is the goal of this problem: to get people thinking with power series.
As always my answer is on the book site I set up. The comments below are just for discussing the problem / people to post their answers.
Tristan and his friend Joe are playing a game of catch. They are fifty feet apart from each other and tossing the ball back and forth at approximately seventy-five feet per second. Two thirds of a second elapse between each of the throws of the ball and the corresponding catch. During that time the ball travels one hundred and sixty seven feet. How is this possible?
Ok, so my first attempt at a lateral thinking problem was a bit of a disaster. Konrad and Jason both were not happy with it and I am trying to figure out a way to re-work it so that the problem is not trying to figure out what the heck I am thinking!
Apparently since I want to publish I need to hold off on posting answers here for copy write reasons. So I will post the answers on a separate site and use comments here as a way to discuss the problems.
So I hate “lateral thinking” problems. Seriously. They usually seem to be more of an exercise in thinking about how other people think – normally an interesting if not enjoyable topic – but somehow sucked dry of the positives of the experience. Never thought much about why I feel that way but I will certainly give it more thought now that I need to write some for my puzzle / math book.
So at some stage I need a tutorial on how to approach these sorts of problems algorithmically – to that end I wrote up this example.
A problem I found online for “lateral thinking” was stated as:
Problem: “How can you throw a ball such that it always comes back to you?”
I am paraphrasing the problem but this restatement has the critical property of the original – at least one answer that is obvious from reading it. Answer: “Throw the ball straight up!”.
These problems always have a basis set of assumptions. For example that the person is in a gravity field or on Earth, that that they can’t throw the ball at escape velocity. I broke down some examples below that have easy graphical representations of the problem. The graphics should help with the book and relaying understanding.
Book notes: examples on breaking down lateral thinking
So I already noted this correction in the previous post – but this is what they find a repeated / missing element in a collection of values from 1…N-1 should have looked like.
Correction - find a repeated / missing element in set of values 1..(N-1)
Just writing this problem down is making me realize how much I have forgotten. Time to crack open the books. I mean I even forgot my notation for mapping into a set with conditions!
[Update: This is why you dont write down things in the middle of the night… the sum I wrote up and *did not notice* is totally wrong. I also made an error on another problem I wrote down. Anyway, the error here is that the sum from 1 to N-1 does match with [((N-1)+1) + ((N-2)+2) + …] becomming N^2/2 not N/2. I just forgot to write down N^2, got distracted, then came back and finished it off that way. Crazy. Yet again I need to never do math after midnight. Anyway ignore what I wrote up below – essentially you have the same solution but the sum you compare with is N^2/2 for a list from 1…N-1 in value. Not going to delete the post as this makes a good example of a simple mistake that can even seem resonable if you dont doubble check your finial result! So might stick this in the book as an example of what NOT to do. ;-) ]
So when preparing for my Amazon interview I brushed back up on computer science as opposed to nocking the rust off my coding skills (read getting back to where I could code my way out of a paper bag). Anyway one of the things I ran across when prepping for the interview was a numeric sum problem. Basically you are given a list of numbers ranging in value from 1 to N, in unsorted order. One of the numbers is either missing or duplicated – write a number to find the function.
( Note: The above is totally wrong, and is an example of an error! See my update at the top of the post for the real answer!)
One solution is to sort the list and then search it for the missing / repeated value. Should be able to do this in O(nlogn) time. I need to check but I *think* a radix sort would work here for N time. A much better solution is to remember the series equivalent of 1…N. Then you can sum the numbers you were given, compare that number with the series equivalent, and just know the answer. Since it takes only one pass over the list it works in O(N) time too.
This is another one from hard to solve brain teasers. It is just another algebra story problem that I have no idea why the included in the book. You just write down what they tell you as equations and end up with two linear equations with two unknowns – so it is fairly straightforward to solve . I do like the way they present the information here as it acts as a good example problem for introducing the idea of graphing or drawing a problem in order help set up equations to solve it.
It would still be a fairly simple problem to solve, but making the problem 3 equations in two unknowns, and using the extra information to nail one of several solutions might make it more fun. So one like this should go in the book.
Good example on graphically representing a story problem
So this was problem 5 from hard to solve brain teasers. I am working through the book and this problem is representative of about 60-70% of the problems which are just annoying. This would not be a hard problem in a 6th or 7th grade algebra course. I am fairly sure the book is targeted at adults – so are these problems hard because you are supposed to not remember junior high? Ugh.
The problem:
“Two cars start from point A at the same time and drive around a circuit more than one mile in length. While they are driving laps around the circuit, each car must maintain a steady speed. SInce one car is faster than the other, one car will pass the other at certain points. The first pass occurs 150 yards from point A.
At what distance from A will one car pass the other again?”
Irregular CIrcuit
The thing I don’t like about this problem is it really is just a story problem where you just get the answer reading it. How? Because the problem lacks enough information to have any other way to set it up. You either get that starting at 0 and ending at A away from 0 means the second pass will be 2A away from the origin as long as the track is longer than 4A in length.
This could be turned into an interesting problem if the speed around the track was not constant in such a way that for the Nth lap A would be in the lead, for the Mth B would be in the lead. Then knowing who wins the race would require knowing the track length. If that was not given but was instead easily calculable the way to solve the problem would not jump out so much. I will have to try and write up a version like that for the book.
Update: Interesting – the problem directly before this one in the book is the same problem class I was proposing to modify this too. I wonder if the author was thinking that when clustering these problems of if that was just on my mind since I just solved that problem two hours ago or so. Huh.
So after that last problem I got to thinking that the area between a circle and its bounding or contained square would be potentially useful to know. Really. It just seemed cool. Only problem is that every use for it I could think of I could think of other ways to get there. Grrrr. Is this really cool – or is it the mathematical equivalent of carrying around a small dog in a shoulder bag?
Finally I have some time to work through daily problems again. Worked this one last week and forgot to write it up.
When I saw this problem I just saw the trigonometry solution and it blinded me to the symmetry. I guess math is good at showing how what we already know can blind us to the better paths.
So why do mathematicians suck? Well that seemed politer for Konrad than saying he sucks – I worked on this problem for a while and just did not see it. He saw the answer before I was finished describing the problem. So I guess I am the one who sucks, – apologies to mathematicians everywhere.
Area of a square - inside a circle - inside a square
So one of the projects that has gotten starved for time lately is working on my grandfathers tool box. My father was, among other things, a carpenter. He worked pretty much right up until he went into the hospital and unfortunately the tools he had at his last job site were “lost”. So when my father recently gave me all of my grandfathers tools that he had it was an interesting collection. The centerpiece of which was two of my grandfathers old toolboxes – including his large carpenters tool box.
Largely empty of tools – the toolbox still had the original wooden dividers my grandfather had made and was filled with an odd assortment of tools (e.g. braces, bits, a crow bar). Now that my dad is retired he also gave me some of his tools and I decided to make new dividers combined with the tools I had gotten from my father re-stock the toolbox.
After taking the dividers in and out of the tool box a dozen times to confirm fit after each modification I decided to make a scale cardboard version of the sides of the toolbox so I could work on the toolbox like one of the sides was removed. Below you can see the tools lid out for test fitting.
Cardboard prototype to test layout
In case you are wondering why it is so much work to just stuff a tool box – maybe it would help to understand the requirements. The toolbox is 32 inches by 9 inches and eleven inches tall at its peak. The top is angled so it is not quite as large as that would make it sound. My goal is to be able to pick the toolbox up and comfortably carry it with one hand. So I am putting an absolute limit of 50lbs on its weight when full of tools. I would also like to be able to get at commonly used items without having to move another tool, or at worst having only to move one other tool.
Here is what I have in the toolbox so far…
2 pannel saws – one cross cut and one rip
Backsaw – dovetail saw filed crosscut
A hacksaw
A coping saw
A pad saw
Crowbar
Cats paw (small pyrbar)
Tiny cats paw
Try square
Speed square
Adjustable square
Framing square (leg of which sticks up out of box)
Large razor blade knife (with extra blades)
Several small razorblade pen knives (with extra blades)
Marking knife (for dovetails)
Carpenters pencils
Folding carpenters ruler
Tape measure
12 inch steel – cork backed ruler
Zippo lighter
Pencils
Notepad
Calculator
Plum-bob
Needle files
Level
Wallet for extra plane blades and extra blades for the different planes
2 1” chisels and 1 1/4 inch “beater” chisel
Angled marking gauge
Combination spoke-shave (round and flat)
Sharpening honing stone
Small bottle of honing oil
Hand scrapers in leather envelope
Small bull nose plane
That sounds like a lot but the crowbar and cats paws go in the small saw till at the back of the box, and the right front half of the box forms an open tool well that got filled with specialty holders. So things can get packed fairly tight and yet be easy to get out without tools banging together.
I still need to finish holders but I know where the following tools will go in the box
Diamond needle files
Collection of dentil picks
Assortment of small sandpaper squares
Telescoping mirror
Telescoping magnet
String level
Plastic impact mallet (for chisels, and “persuading” things)
Hammer
Low angle Jack plane
Skew block plane
Small Rabbet plane
Screwdriver for plane adjustment
Tiny brass hammer for blade adjustment
A Stanley 60 1/2adjustable mouth block plane
Needle nose pliers
Lineman’s “dikes”
Small wire cutters
Locking pliers
Chalk snap-line
Even with all that there is still some room to work with. My problem is that I still need to pack in the rest of these tools / items.
Torpedo level
Small hand mirror
Several neodymium magnets
Several 1 and 3 inch C clamps
String for line level and plumb bob.
Several bevel edge socket chisels for dovetails
Extra blades for coping saw and hack saw
Glass cutter
Push drill and bits
Set of good screwdrivers
Earplugs
Eye protection
Rag
Leather work gloves
Small thing of super glue
Needles and thread
Simple first aid – band aids, antiseptic cream, ibuprofen packet
My main problem, I think, will be figuring out how to get the screwdrivers in the right side tool well – and easily accessible – without making it hard to access they other tools.
Below you can see me laying out my planes. They will sit in a small wooden pull out drawer. The pulls out portions are going to be made from cherry – while inside the box I am using mostly poplar and a decent grade of plywood.
Planning layout for plane pullout box
The planes above are a Lie-Nielsen sckew block plane, low angled jack plane, and rabbet plane. The jack is amazing in that it can be set for rough or finishing work. I have a spare blade for it and also got a 90 degree scraper blade to be able to use it for finishing. The skew block plane is useful for both a block plane or for use in dadoes. I have a set of side rabbet planes and a place reserved for them – but I am still on the fence about adding them.
Left side storage area
Here you can see the left side of the box. There is a small well area on the bottom of the box. That is where the oil-stone and plumb bob will live. This is where I am thinking the chalk snap line will go. Along the side of the space is a pocket for the scrapers and a bin for pencils and pens. The reason there is about a quarter inch of dead space along the side of the toolbox is that is where one leg of the large framing square sits when in the box. It is a bit of a hack but I plan on sewing an envelope to hold an assortment of pieces of sand paper. That envelope will sit on top of the carpenters square in this dead space.
The drawer with the plane box sits atop the left side storage well. I should also be able to have another box atop that and then room for a smaller pull out tool drawer at the top of the box. The layout was bugging me so I went and made a cardboard mockup of that top drawer. That way I can build it and the plane drawer and then make the center box to fit. Not quite optimal but I am fairly sure I will be able to fit the chisels I want – as well as the sewing kid and some other tools in those center boxes.
Cardboard prototype of left side top lift out tool drawer
Here is what the right side tool well looks like – if the back of the toolbox were removed. This has proved the most useful attribute of the cardboard mockups of the box’s sides – being able to visualize the insides as if I had x-ray vision. It is kind of like making simple test cases when developing software.
Right side tool well and storage area
So I am currently laying out a small set of holders to go next to the chisels, but on the other side of the coping saw. Should look something like the picture below. I am still trying to figure out if I want to have dividers between the tools. On the one hand it would keep the tools from rattling around but it would mean I would have to have *those* tools and could not swap tools in and out as needed. Still on the fence about how to do this.
Planning layout for new tool well in right side tool well
