Math Journeys, Volume 1
A Rich, Little Problem, Part 1
If you’ve browsed the Puzzle Mall at our site, you may have come upon the problem entitled “Reversal of Four-tune! A Rich, Little Problem.”
This math journey will show just how rich a simple problem can be. We’ll stay mostly in the arithmetic sphere, although blanket solutions, called generalizations, are what math people always look for, and they often require some algebra.
If you care to travel along, you’ll do part of the work, refresh your skills, and maybe experience the solver’s exhilaration a few times. Stay curious, ask questions, and keep searching, and you might experience the discoverer’s exhilaration, an even bigger high.
Let’s begin to solve our little problem. Take in the big picture – it often saves work. Estimate – if R in the factor is 3 or more, would we have a 4-digit product? Think about parity – is the product UTSR odd or even? Without using any pencil graphite, we’ve established that R is 2, and that means U is 8 or possibly 9, but 4 x 9 = 36, and that would make R = 6. So we know that U is 8 and R is 2. Solving for S and T involves a bit of trial and error. If your multiplication is rusty then use a pencil rather than a calculator.
After you’ve solved it, think about related avenues to explore. There are many. Here are some questions that occurred to me right away.
1. Do other multipliers reverse a four-digit number?
2. Are there 3-digit numbers that reverse?
I’ll stop there to let you think of some questions of your own ..., and I’ll drop a little teaser on you. Would the fact that 1.5 x 4356 = 6534 have any connection with our investigation?
To answer these questions and more requires much digging, and so we’ll spread it out over a few months – that’s why I call it a journey.In the meantime, you can finish solving RSTU, and work on the other two questions above. Here’s a start on them:
1. The only other single-digit multiplier that reverses a 4-digit factor is 9. Can you find the factor?
2. There is some checking involved in looking for 3-digit examples, but not as much as you think. For example, 9 x ABC = CBA, would necessitate an A of one, and that means C = 9. Does it work? Another example to quickly dispose of is 5 x DEF = FED. D in the product cannot be zero, and so D is 5, but that makes the factor too big because 5 times 500 is a four-digit product. The search is on!
Next month – answers to the above, a new trick, and the “teaser” joins the family.
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