I just really liked this movie quote.
Movie: Open Range
“…a man’s trust is a valuable thing – you don’t want to loose it for a handful of cards.”
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This problem kind of bugs me. Here is the problem and my first step:
So I like my first step – but then my way of solving {a(a-1)(a-2)(a-3) = 120} is wonky. I want to expand it and use the quadratic equation – which is not slick enough to be what they are after. I am not seeing synthetic division, or any usual suspects – so I think I am missing something obvious.
Rick, someone I work with, looked at the problem and immediately said you could tell that the solution needed to be divisible by 5, which happens to be a valid solution here. However I don’t believe that is always the case though.
My way to get the solution shows a solution of {-2,5}. Work shown here.
This is another example of a substitution problem. Normally I would have just worked it out long hand, expanding thing and solving. That’s a lot of work and frankly kind of a foolish way to attack the problem looking at it now. Since I started looking for substitutions first – the problem collapses to simple in one substitution. Unfortunately its still fairly un-elegant – so I don’t think this is the solution they book is looking for.
The problem is stated as: “Solve: (6x+28)^1/3 – (6x-28)^1/3 = 2”
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Mathematical Quickies No.205
I kind of hate problems where the solution is to find the most elegant answer possible. This problem is an excellent example of why. My solution (here), solves the problem as stated – but I took a grinder solution just wearing the problem away. My gut says that there is no way that this is the solution they want – however I do solve it with multiple applications of the quadratic equation, which the problem framing hinted at, so I am posting it for now.
Knowing that this is not the solution they wanted, I want to go look at the back of the book to confirm that is the case, and see what they were after. The problem with looking though is that I ruin the problem. So, I pretty much never want to look – hence my dislike of problems I where I can not prove I have the right solution. They all seem to be asking me to prove that someone smarter than I am could not solve the problem in a more elegant manner, but since the person in question is by definition smarter than I am – well – it seems a might ridiculous.
Here we have the problem:
“Solve this equation using nothing higher tan quadratic equations:
X = Sqrt( (X-(1/X)) + Sqrt(1-(1/X)) )”
So for some reason I never started looking for substitutions when solving equations. It was certainly something I learned to do when analyzing circuits, but when I see a math problem I never started looking for substitutions that might simplify the problem. Until recently.
Looking at, I think this problem, it “just clicked” – and I started looking for substitutions. Then used them to nock out answers for the next half dozen puzzle problems I tackled. Weird, since I am not doing anything I did not know before, but I just started looking at problems differently.
This one is fairly straight forward:
Solve: (x-a)/b + (x-b)/a = b/(x-a) + a/(x-b)
It becomes way easier to solve after a simple substitution.
My Solution.
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Gebhard Solutions training knives
Gebhard Solutions has some interesting training blades for small folders. I want make one like this but I think I will make a pattern from scratch. Mostly as a learning experience.
Finally got a chance to take pictures of the second loveless prototype. First thoughts are that the white under wrap with black over wrap did not look as nice as I was hoping.
I still want to try a flat under wrap – which should let me space out the over wrap a little better, and father apart. Regardless the top side insert was about 4-4.5mm too long, leading to the notch of empty space by the lanyard hole.
I do think this blade fits better in my hand. With the insets the wraps make the handle about 4mm thinner. I think I could take it down another 2mm and then weight the handle to shift the center of mass back into the hand. Before I do any major reshaping of the design I think I will make the next blade from steel and see how that impacts things.
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Testing blade strikes
So the second bit of feedback I got from my knife fighter friend was that her training blades always get banged up in knife to knife fighting. I made the blades form 6061 aluminum, and was curious how it would hold up.
I mounted some scrap stock in a vise and did some slash cuts on it, then let my friend do the same. I marked the blades with sharpie so that the damage would show up better.
*Ahem* I looked like I was waving a tiny baseball bat around with one hand and her strokes looked like she was performing an appendectomy. The result of both our cuts though was gouges on the blade. So while 6061 Aluminum is probably fine for training knife defenses, I think I will want to make the final training blades from tool steel so that they can also be used in knife to knife training.
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Rethinking the handle wrap
This blade design is inspired by the keen-edge blades 9 inch loveless. After the handle felt fat in my hand, I took a second look at their pattern. I double wrapped my handle – so right off the bat my dimensions were bigger than theirs by 6mm. So I figured I would cut the same knife but with either side of the handle inset by 3mm on each side. (Inset shown with the red arrows).
I also added a small guard on the back side of the blade opposite to the finger guard. The idea is the extra guards and insets would simultaneously make the blade thinner even after I added a double wrap, and help keep the under wrap from sliding around on the handle over time. All in all I think I want to play with this idea more. I like making the handle feel thicker with a double wrap – but without its feeling wider.
Unfortunately I dropped the knife off at the dojo for people to play with before taking pictures of the new design.
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First prototype in hand
So I got some time to play in the shop this weekend. Once I adjusted g-code generation to just pencil line the profile of the blades I got the cut time for both blades down to 17 minutes. That was going super slow at 5IPM, since I did not want to drive Z faster and the g-code generator was not playing nice if I tried to have the axis at different top speeds.
For this first blade I worked off the loveless 9 point hunter. I really liked the lines on the keen-edge knives pattern for the loveless. In the end it seems a little small in my hand. Maybe I got the pattern wrong but the finger guard presses into my hand.
I did both the handle wrap and the over wrap with 3mm para cord – so the handle also feels a bit thick to me. Interestingly that bulk made it seem handle heavy to me, but as soon as I placed the knife in a friends hand who is a knife fighter her immediate comment was that it was blade heavy. Adding a false blade profile should help – but this drove home that I am going to need to get a lot of feedback from blade people to make this project work.
The reverse grip feels a little less “fat” in my hand, but I don’t know now how to improve it. A little stumped here.