Question

Dear Mitch,

You recently posted some interesting problem-solving questions, which you promised to answer before Valentine's Day so that any teacher using them would feel comfortable that the answer he arrived at is, in fact, the one that is correct.

Valentine's Day is the day after tomorrow...

So, would you please be kind enough to provide the answer to the problem about the three people sharing a hotel room to see the Valentine's Day parade, and would you also provide the answer to the one which even you acknowledge as very difficult, even for high school students. (That one was the one about the gold coin company that makes 100 gram coins ...).

Thank you,

Mrs. Packard,

Ninth Grade Mathematics Teacher,

Independent School, New Jersey

Answer

Dear. Mrs. Packard,

Or course!

First, just so we have the questions and answers in the same place for the sake of easy locating and printing out, here once again are the questions:

- QUESTION: GOLD COIN COMPANY

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 that each equals exactly 100 grams, no more and no less than that. They are 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", 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.

__$30 HOTEL ROOM AND THE MISSING DOLLAR__

This is a Valentine's Day twist to a classic problem. You do not need to know much math at all, but you do have to try to think mathematically (or logically, or REASONABLY).

To get a good view of the Saint Valentine's Day parade, three friends decide to avoid the crowd and treat themselves to a hotel room right on the town's main avenue so they could stay warm and dry (if it rained) and still enjoy the view from a near-perfect location.

The hotel room is $30 (a Valentine's Day Special), and so the three men agree to chip in $10.00 each.

The bellhop, who is also the hotel's manager, shows them to their room and collects their money. He wishes them a pleasant afternoon and goes back into the elevator. But as it begins to descend, the man realizes that the room he gave the men was really only a $25 room. He feels he should do the right thing and return the five dollars to them even though they do not know it is on sale. So he rides the elevator back up to give them their refund, and as the doors reopen in front of their room, he realizes he does not know how to split $5 evenly between three people. (He tells himself he was absent that day in math, because he was pulled out for an orthodontist's appointment). He begins to get nervous, does not want to embarrass himself, and thinks quickly. Then he decides what to do.

He has one-dollar bills in his wallet, so he'll tell the men the room is reduced from the thirty, but he'll only tell them it's reduced by three dollars. The room is twenty seven dollars, instead of thirty, he'll tell them, giving each one dollar back. And he will keep the two dollars for himself as a little tip (that's stealing, but on Valentine's Day, we won't focus on that part. After all, he figures, the men will be happy they're each getting a dollar back, making each of their shares 9 dollars each, which is a pretty good savings. And the best part is he'll have two extra dollars for himself, which he'll use to buy a math book and brush up on basic facts. . . Either that, or candy, but he would not have to decide right away).

Indeed, he gives each of the three men one dollar back, and they thank him. The room only cost each one of them nine dollars instead of 10. After all, it makes sense:

9 x 3 = 27

And, of course, that $27 + the two dollars in the bellhop's pocket makes $29. The only mystery is this: They started with $30 and now altogether have $29.

*Where is the missing dollar?*

** ANSWER TO GOLD COIN QUESTION**:

You, the detective, take a different number of coins from each employee, such as 1 coin from employee # 1,

2 coins from employee # 2,

3 coins from employee # 3,

etc.

(Note, the problem never states how many employees there are, so if there had only been 3, then you would be done with step one. On the other hand, if there are 50 employees, you would not be done with step one until you work your way up to 50 coins from employee # 50.)

Next you calculate how many grams that total collection should weigh if no employee was stealing one gram of gold per coin. For example, if there were only three employees, then 1 coin from worker # 1, 2 from worker # 2, and 3 from worker # 3, would equal 6 coins, which should have a total weight of 600 grams.

Next, you toss/place them all on the scale at once. (A basically weightless plastic grocery bag might make this more convenient but is unnecessary.)

Next you read the scale. And.... the number of grams it shows will be a certain number less than the number you calculated. Since each of the thief's coins were exactly one gram less than the correct amount, the difference between what it should weigh and what it does weigh is, of course, the number of coins that are made by the thief. And that number is the number you've assigned to the worker who is stealing gold. Example: three grams less than it should be, and it is worker number three.

Hope that helps!

** ANSWER TO THE $30 ROOM AND THE MISSING DOLLAR QUESTION**:

This one is deceptive in the way it seems so simple and straightforward and then, out of nowhere, ends illogically.

It has to do with the way the wrong operation is switched at the end and numbers are put back together in a way that is not mathematically equivalent to the way they should be.

It begins with 30 dollars, formed by each of three people contributing 10 dollars.

Then it turns out that they are only supposed to be charged 25 dollars for the room, so five dollars are to be returned to them. Instead, the bellhop keeps two dollars and gives them each one dollar. This means they each paid ten dollars but when receiving the one dollar rebate ended up paying nine dollars each. 9 x 3 = 27. "PLUS" the extra two dollars kept by the dishonest bellhop. 27 + 2 = 29. WHAT?????

So, we look back at how that happened:

When the three dollars were redistributed among the three guests, it is true that they each paid nine dollars. It is also true that they then paid a combined 27 dollars since 9 x 3 is indeed 27. Finally, there are the two dollars in the bellhop's pocket. They should NOT be ADDED to the 27. THEY SHOULD BE SUBTRACTED FROM THE 27, because the twenty-five dollar price is costing 2 dollars too much, and the stolen two dollars is the reason. They did indeed pay 25 PLUS the two dollars stolen which they do not know of. 27 – 2 = 25, which is the corrected price. In other words, 25 + the 3 that they got back + 2 that the bellhop kept = the original $30.

Hope this helps,

Happy Valentine's Day,

Mitch