Question
Last week's question was as follows::
Dear Mitch,
I am a middle school teacher on Long Island. This June I celebrated my twentieth year in teaching in my current school.
A few years ago, you were hired as the guest lecturer for our district's annual Teacher Inspiration Week ("TIW"), which marks the important beginning of our 4-day faculty week. In those days you were best known for the power of your guest lectures, and the fact that you somehow made each lecture unique. In light of that, you might not recall the speech you gave to our faculty that day. I sure do! Our entire math department changed the way it taught several fundamental topics. You influenced even some of the old-timers who can be very 'set' in their ways and protective of the way they do things. Here's my question to you, sir: After you presented a comprehensive collection of "hands on" methods using materials that most of us had never even considered using, and some of which I have used each year since, what I found just as intriguing was the last quarter or so of your lecture which you set off by announcing that we were now going to switch gears and demonstrate how powerful the opposing style can be; no hands-on exploring, just storytelling.
Our district was exploring many of the acclaimed math story books, particularly for younger students, but you took an approach that is almost never taken: rather than read from a book written for that purpose, you told a few stories which were clearly true about ordinary experiences and how math or problem-solving skills made a huge difference to everyone in such a situation.
And there was one you described that had something to do with a small closet or cabinet you had built in your home that had some kind of problem that became clear the moment it was finished. And that's where I'm stuck. I recall enjoying the story, and, of course the problem, but I can't recall how you came up with a set of numbers that work. And as lovely and as funny as the tale was, without a set of numbers that work even half as well as the ones you had it doesn't have the same impact.
Sincerely,
Mrs. McFree
Answer
(PART TWO FOLLOWS DIRECTLY AFTER PART ONE, SO IF YOU RECALL LAST WEEK'S BEGINNING, THEN SCROLL DOWN TO THE WORDS 'PART TWO'.)
Dear Mrs. McFree,
I know the one you're thinking of!
You were correct, it was a closet in the kitchen.
We had a cupboard built underneath a staircase to store canned foods and other small items.
Unless we were willing to have major reconstruction done to the house (we were not), it was to have a height limit of five feet. (divided vertically into two matching sides, like a narrow refrigerator/freezer combo that's split vertically. It was only six inches deep because the whole point was to build it flush with the wall. It was designed with adjustable shelves with standard clips, and for each side I measured a few favorite products and after trying a few different arrangements, settled on using 8 shelves on each side, some allowing for tall items, and some not. (A Heinz ketchup bottle would be the tallest, and mustard, for example, would be one of the shorter items). The cupboard was sealed by two plane matching doors each a couple of feet wide. Like this:
_____________________ __________________________
_____________________ __________________________
_____________________ __________________________
_____________________ _____________________________
_____________________ _____________________________
______________________ _____________________________
_______________________ ____________________________
______________________ ______________________________
I was pleased with it, and thought it would be one of the most useful changes we'd had made to the house that year. But there was a problem.
I was informed that although the closet was a delightful idea, unfortunately, as luck would have it, it turned out to be just a bit too small. And since the items that didn't fit into it would have to be stored in the basement, the closet failed in relieving the person cooking from having to venture down there. The closet had to be enlarged. Two carpenters were called in to give opinions on enlarging the closet. They both agreed that the obvious solution would be to add six additional inches, vertically. It would reach inside one of the steps but would cause no problem because the strength of the step would be restored by a technique involving adding steel supports to reinforce the six inches of wood which would have to be cut away. No problem.
Estimates:
Carpenter number one: $760.00 (remake doors to fit enlarged front, recut wall and additional support to top, painting, finishing, and if we were to give him an immediate Yes, we would get a commitment that he'd do his best to start the project by August 1st. The magnetic calendar on the refrigerator indicated that we were in December.
Carpenter number 2: Not $760.00. No, nothing like that.
$860.00.
He could probably get it done in a month, but don't quote him on that.
But by all means, he said, we should feel free to quote him on the price. $860.00.
$860.00.
QUOTE:
"$860.00."
I thought of all the benefits of keeping certain foods in a basement environment.
"Like wine in a wine cellar," I said.
To which someone passing behind me replied, "But you don't drink wine."
"Still, it's nice to have around at holiday time in case guests show up."
"But you don't like guests."
"Not when they ask for wine, that's true."
She nodded. "Eight-hundred sixty dollars" she mumbled softly.
"I'm thinking there's got to be a way to do this without so much energy."
"Energy?" she said, "You mean the eight-hundred, sixty dollars."
"Energy," I repeated. Then I mumbled, "And yes, eight-hundred-sixty is some number."
"Why do you say that -- is it a special math number?"
"No, it's just some number."
"Oh."
Later, when I was the only person awake, I sat down in front of the closet. I leaned forward and opened the doors and thought.
Often, when trying to solve a problem, the process could involve hundreds of different attempts over a ten year span. More often, though, hundreds of different ideas are attempted and discarded within a hundred minutes. And that, of course, is less than two hours (which is 120 minutes). But to solve a real-world problem and implement the plan, which in a case like this might require an experienced carpenter, such periods of time would be considered miraculous. And, in addition to the cost in time, implementing a change would almost certainly require accepting a period of noise and mess -- from cutting the wall open to accommodate the taller doors, to getting the small ones into a dumpster and hauled away. And then there's sawdust...
In fact, this puzzle was so perplexing that I had to stare straight into it for just a few seconds shy of 175 minutes before my entire skull arched into a smile. 175 minutes! And that's almost three hours (which would be 180 minutes). But, by the end of that period, the problem was not only solved, but the solution was implemented and the last bit of sawdust swept up. (There wasn't much to sweep, and the best part is that the following morning when the sun came up and shined into the kitchen the problem appeared to be right where they'd left it, though now without the merciful cover of night and you couldn't take a break from its newly painted trim, making it even harder to ignore.) Fortunately, though, the instant its doors were
opened -----------------
the solution presented itself.
......STAY ......... TUNED...........
NEXT WEEK, SAME MATH TIME.... SAME MATH CHANNEL....
PART TWO:
Okay.
Welcome back!
As is the case for any such 'problem', there are many solutions. I usually settle for one good one.
Recall: There were 8 shelves on each side of the 5-foot tall cabinet. That means there are 16 shelves all together. Apparently, we needed 17.
Again, the closet had an interior that had a height of five feet. Since one foot equals 12 inches, 5 feet equals sixty inches. Therefore, regardless of the arrangement (with some shelves leaving tall spaces and others short spaces), they averaged 7 1/2 inches of vertical shelf space per shelf, as 60 inches divided by 8 shelves equals 71/2 inches.
Correct?
NO!
NO, No, NO!!!!!
That is not correct.
Why?
Because this measurement does not account for the piece of space taken up by the thickness of the wood of each shelf.
I usually try to figure out solutions by estimating and rounding first, but here I sensed the whole thing would come down to sums of very small measurements. So, using a ruler, I measured the width of each shelf: 3/4 inch plywood, which was easy to measure, even though the edges were covered with a fancy-looking eighth inch veneer (apparently, "looks matter" even for the inside of a cupboard which is hidden 99% of the time by the doors that close on them.) If the issue was not that its appearance mattered, one would have to consider the idea that the person building the cabinet made the mistake of thinking one of our goals in having it built was to increase the quantity of his work and/or the price of materials. And since that idea is absurd, we must logically conclude that cans of soup cannot be expected to stay all night locked in a dark cabinet being touched by unrefined shelves.
I hadn't thought about unnecessary wastage as much as I sometimes do because the veneer did in fact look quite nice. However, when I noticed the pile of scraps accumulated during the building of the closet, I discovered that the veneer selected was some kind of paneling, rather than a 3/4 inch roll of wood one would have guessed gets glued on.
Okay. I did not know much about different kinds of wood, but I did know enough to see that this paneling was not cheap stuff. Fine. 8 shelves, each 3/4 inch thick, all together equals six inches exactly.
Note: Woods come in various sizes and the way it typically works is the person having the item built pays for every bit of wood that is purchased, so using a one-inch piece of wood that comes in eight-foot long sheets requires the purchase of the entire eight-foot board. This is true, despite the frequent fact that the majority of the wood purchased may end up in the scrap pile.
I looked at the paneling leaning against the wall. It looked right back at me and – if I didn't know better – I'd say it winked. I winked back and then decided to alleviate the pain it must have been experiencing from feeling useless, cast aside, and, eventually, neglected.
Using chalk I traced one of the 3/4 inch shelves onto the dense paneling. Then I traced it 15 more times. My dream was to trace it 16 more times, which would have given us the extra shelf, but I'm too superstitious and did not wish to count chickens or shelves before they hatched. This is what I was thinking: 16 shelves x 1/8th inch thick paneling equals 2 inches of space that would be filled with the solid wood of the shelves (compared to the previous set of 16 shelves, each 3/4 inch thick, which totaled 12 inches of solid wood. So we would gain ten inches of free space. From that, we would have to subtract the 1/8th inch of wood from our new shelf, and we would get 9 7/8 inches of new free space.
Oh, I thought, when everything is revealed, how the salsa will dance with joy!
I found my 50 foot extension cord and my car keys. I started my car and positioned it so that its headlights would face the back corner of our backyard. I plugged in the carpenter's saw which was to be picked up the following day. Yes, I borrowed the man's saw without asking. I did pick up the telephone to call him, but it was 3 a.m. and decided he would not be as inclined to say yes as he would be the following day after it had occurred. (Besides, I once read that it is not polite to call people in the middle of the night to request favors.) I figured this way: during his time at our house, he borrowed our toilet (twice) our floor (constantly to stand on and scuff up, rather than figure out some way to remain hovering above its surface), he borrowed our sink to wash his hands (four times), and our driveway to get to our front door (twice on each day he worked). I set a timer for one hour so that I would be truthful the following day when I would ask him how much I owed him for one hour of the saw's use. If the fee seemed high and was accompanied by an attitude that lacked the kind of generosity that I believe is important in today's crowded world, I would produce an itemized bill for all the things he borrowed from us during his 'work time', and so we could work things out more precisely.
Far from the house, so that I woke no one, I cut all the shelves in under forty minutes. Then, figuring I'd chance it, I cut the final shelf, deciding I was willing to trust the mathematics enough to count chickens and shelves before the hatching .I returned to the house and set everything in place.
I was careful to return everything to the cupboard and leave one shelf, which now had the largest space above it, completely empty.
Empty, that is, except for a little sign I made that stood on it.
The sign read:
"What about me?"
And the three-quarter-inch shelves that no we no longer required?
I built a small bookcase-type cabinet on the basement wall just a few steps down from the kitchen. This way, if, in the future, our appetites were to grow further, we would have additional space to store items, and the person cooking would still not have to venture a full 12 feet below the surface of the earth to locate them.
Hope this helps,
Mitch