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Question

Dear Mitch,

A few days ago you presented a classic world problem that I was excited to finally see in print because as a math teacher myself I too had heard it many times over the years. (I' m referring to the one about the three men splitting a $30 hotel room which later turns out to be only twenty five dollars, resulting in a story in which one of the thirty dollars mysteriously disappears.) My favorite part about your presentation was how you rewrote the problem for Valentine's Day and somehow, in doing so, made it feel as new as it was when I first heard it all those years ago. Anyway, if I'm not mistaken, you put the problem up about a week ago, and I believe you mentioned you would present the answer in a day or two. It's been so long since I first heard the problem that I honestly cannot recall where the dollar went, so, as you can imagine, I am eagerly awaiting your wind up to it. Did I miss it or is it yet to come?

Sincerely,

Dr. P. Carson

American Ex-patriot Teaching in Paris

Answer

Dear Dr. Carson,

You did not miss it, but this problem has elicited more responses from readers than we've received on any other question posted in the last four months, so we've decided to let the problem solving continue for at least another few days.

However, thank you for asking, because your question someone brought to mind another famous classic, one which has been among my favorites since I first heard it as a child, and now I am going to share my own Valentine's Day version of it for further pondering. I hope you and your class find it interesting, and well... Good luck!

There is a gold coin company which makes, as you might guess, gold coins.

Each coin is considered a work of art, and so the factory is not set up in an assembly-line fashion, but instead is arranged so that each employee works alone and makes each coin from start to finish. There are certain requirements within which they must make their coins, regarding such things as size and shape (circular), but the most important of the specifications is that each finished coin must weigh exactly 100 grams. The way that the company's owner makes the weight aspect so easy to manage that the artists do not have to even think about it is this: The raw material (the lumps of gold) that are to be turned into coins are carefully weighed and checked to assure each equals exactly 100 grams, no more and no less than that, and arranged in evenly spaced groupings so that when a worker is ready to begin he or she simply has to grab a lump and begin the task.

HOWEVER, as Valentine's Day begins to approach, and the demand for these romantic coins rise, a problem is discovered: One of the employees is stealing from the company.

The thief is one of the artists who makes coins, and the way he or she is stealing is by somehow managing to remove one gram of gold from each lump during his or her coin-making process, pocketing that gram, and turning out 99 gram coins instead of the 100 gram coins which everyone else is dutifully making. The problem is that it is impossible to tell by examining one of these coins (or even examining more than one) whether it is a full 100 gram coin or the slightly lighter 99 gram version.

Task: You are hired as the detective. Your job is to find the guilty employee. HOWEVER, it is a very old and traditional family-based company and the owner who hires you informs you that it is of the utmost importance not to ask employees questions that could embarrass them, and it is just as important that the whole mystery be solved quickly and with just one 'move'. Specifically, you are given a spring scale (that is the kind of scale that people often have in their homes to weigh themselves, and to do so the person simply stands on its surface and waits until the 'needle' or line steadies at one number.) So, it is NOT the other kind of scale, known as a "balance scale", and is the type that has a plate or tray or bucket suspended on each end of a stick-like piece that pivots on a single point or 'fulcrum'. (These scales, balance scales, usually look like the see-saws or teeter-totters at a children's playground, and are more often used in science classes than spring scales.)

You are informed that you are allowed to use the spring scale only once, meaning only to make one measurement, BUT that measurement can be as large or as small as you wish. So, for example, if you decide you would like to weigh one coin, you can toss that coin on the scale and that is your one and only measurement. Alternatively, if you decide you would like to collect a hundred coins and throw them on the scale, the measurement they produce when the line settles on some number may be very large, but it is still your one and only measurement. To reiterate that point, you may take as many coins or as few as you wish to weigh, but you are only allowed to place them on the scale in a single move, thereby producing a single weight for you to contemplate.

Question: What single act can you do with coins from this company and a spring scale to guarantee that you will be able to figure out which employee is producing the malnourished coins? Remember, this is an important task, because it would be terrific to catch the thief and have him (or her) fired, removed, and convicted in time for Valentine's Day.

Now hurry!

Hope you and your students find this as interesting as I always have, and have fun!

Hope this helps,

Mitch

P.S. This particular problem is typically considered extremely difficult for high school students to figure out in less than a week's time. But worth a try!