Ernie and the disappointing drill bits

Question

I was driving to Ernie's place a couple of days ago, with Ernie in the passenger seat, when we drove past an old building with a faded sign reading Lar's Tool's. Beneath it, scrawled on the front wall of the shop in white paint, a second sign read Closeing down sale - last day's - 75% of all tool's (clearly Lars' specialty was not grammar or spelling). Ernie, always keen for a bargain, asked me to stop for a quick look. Inside, the gloomy workshop was packed with scarred old tables heaped with an assortment of dusty equipment. Lars (presumably) was the elderly gentleman sitting beside a battered cash-register. Ernie asked me to check out a table labelled Loose Drill bit's while he investigated a pile of antique bench-vices.

I was pleasantly surprised to find that underneath the dust many of the drill bits appeared to be sharp and unused and I was able to sort out two identical sets of seven Acme Industry metal-working drill bits, each consisting of a 1 mm, 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, and 7 mm bit. I carried them over to Lars and paid the "$1 per bit, cash only" that he requested. When I enquired if he had box to put them in Lars told me somewhat gruffly that the place wasn't a box shop so he didn't have any boxes. But he then handed me a roll of duct tape and directed me to a table down the back of the shop that was laden with various off-cuts of metal tubing of assorted materials and dimensions. "Maybe you can put the drills into a couple of those tubes".

After a bit of experimentation I chose two suitably-dimensioned metal tubes, placed a complete set of seven drill bits (sizes 1,2,3...7 mm) into each, and sealed the ends off with duct tape. I went back to Lars and paid for the tubes "10 c each" (the duct tape was free, it seemed) and then returned to the car with Ernie - he hadn't found any other bargains in the shop. Back at Ernie's place I handed him the two tubes. He shook each one, then peeled the duct tape off one end of each tube in turn and looked down at the drill bits inside. "That's interesting.", said Ernie, "One of the tubes has precisely the largest possible internal diameter to be able to hold the set of seven drills such that none of them are able to rattle from side to side. The other tube has precisely the smallest possible internal diameter to be able to hold the set of drills such that at least one drill can rattle from side to side". But when he tipped the drills out onto his work-bench for a closer look things did not appear so promising. "Hmmm," he said, "I can see why the bits were only a dollar each. I am pretty sure they are cheap Acne Imdustry knock-offs of the real thing. They are unlikely to keep a good edge. But thanks for the thought anyway." I was a bit disappointed for letting him down on what I had thought to be such a bargain, but Ernie told me not to worry and that they would be useful for something. So, we had afternoon tea, chatted a bit, and then I headed home for the evening.

I didn't think any more about the day's events until Ernie phoned me with excitement in his voice yesterday morning. "That stuff you got from the tool store - the drills were rubbish, but the thin-walled stainless-steel tubes you packed them in are perfect - exactly the right internal diameters I need for my next experiment". He went on to explain how the tubes were non-standard diameters and he hadn't been able to source similar stock from his usual laboratory suppliers but thought that perhaps there could be some more identical tubes at Lars' place. "They don't have to be exactly identical", he continued, "so long as they have the same internal diameters to the nearest micron". I promised that I would have a look and hung up the phone before realising that Ernie hadn't told me what the two required internal diameters actually were.

I have tried ringing him back but he isn't answering - probably deep in thought in his lab with the phone off the hook. I could keep trying to contact him - Ernie usually leaves his lab within a few days, but I'm concerned that that the shop might have finally closed down permanently if I delay too long. All I can think of is that Ernie's comment regarding how the drills were packed into the tubes might offer a clue regarding their dimensions. Can anybody help?

Answer 1

My job title was Tooling Engineer for a number of years, so I suppose it was fate that kept others from answering this question before I got around to seeing it. Or perhaps lack of CAD software.

The tube with some loose bits:

I'm looking for the smallest diameter container that still allows at least one to move. Ensuring that one bit moves wasn't going to be much of a hassle... you actually have to try hard pretty to get that 1mm bit to not move. All the competitors for smallest look like this, having the 5 largest bits being arranged against the container walls in some order. There aren't too many to check, and this was the winner. Both the 1mm and 2mm bit were loose. The container size was 13.462111mm according to my software, which is 13.462mm to the nearest micron.

The tube without loose bits:

I want to find the largest container that holds all bits without movement. This one was trickier. I played around for while just to understand what it takes to achieve no movement and this was the first one I found which makes for a good example of what we want to see: I drew phantom lines only between circles that touched. These split each circle into sections between points of contact which is handy because to to be rigidly locked in place, no section can be more than 180deg. Next, observe that those lines are forming quadrilaterals in there. Quadrilateral grouping have the advantage of being "malleable" unlike the shape you're probably more likely to make... a triangle. It is counterintuitive that you want malleable quadrilaterals instead of rigid triangles, but the reason comes down to your container wall being so important that having it touch 4 or more bits is basically required and for that happen your bit arrangement needs to be sufficiently squishy. Turns out having one triangle isn't enough to mess with that issue, because my actual answer uses one: I have no idea how to know if it is the best in the mathematical sense, but just looking at it I knew it is at least really good. 2mm and 6mm do not touch, though it looks like they do because the gap is a mere 5 microns. The 3mm bit is very nearly not constrained, but those two contact points are less than 180deg apart (about 173deg). That angle being so big is essentially the reason I stopped my search, assuming I had found the best. You can see how it helps form a big bridge over that void in the lower left. The container diameter is 14.761725mm or 14.762mm to the nearest micron.