The Backward-Forward Method

Unfortunately, little or no time is spent in most math classrooms discussing "heuristics," the art of problem solving. 

This often leaves students grasping at straws, struggling even to know where to begin when staring down an unfriendly, unfamiliar math question.

George Pólya's How to Solve It is a classic on this subject, required reading for all serious math students.

In another classic, How to Read and Do Proofs, author Daniel Solow advances a powerful problem solving approach he calls the “Forward-Backward Method.” 

I’ve found it helpful in my own teaching and mathematical work to reverse the method, first thinking backward from the ultimate goal to various subgoals which, if achieved, would enable direct progress to the original objective.

Whether writing complex proofs or tackling simple algebra problems, this “Backward-Forward” process provides students with a simple yet powerful structure for solving problems.

Just as an archer would prefer to move the target closer, so can a math student make a problem easier by finding a nearer target to shoot at. The next step would be to think further backward, to find another even closer target tied directly to the first, and so on. These subgoals are set by repeatedly asking the same question: “What would I need to know to find that?” And then “What would I need to know to find that?” Subgoals should be written down, to keep the trail clear.

After looking backward as far as possible, it’s time to reason forward from each given fact, with the last subgoal in mind. A different question governs the forward process: “What can I imply from that fact?” And then “What could I imply from that?”

Eventually, forward progress enables us to hit the target. All that’s left is to follow the string of subgoals up the ladder to the desired result.

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Fox, Chicken, Grain

"Fox, Chicken, Grain" is one of my favorite logic puzzles for younger students.

Here are the rules.

A farmer wants to successfully ferry a fox, chicken, and sack of grain across a river. He can take several trips back and forth, drop things off, and pick things up on each side of the river between trips. But he can carry only two things in his small boat, other than himself, at one time.
Unfortunately, the farmer has a few problems.

If left alone together on the same side of the river, the fox will eat the chicken, and likewise, the chicken will eat the grain. But the fox will not eat the grain (and, of course, the chicken will not eat the fox).

How to get everything safely across the river?

I usually use coins of different sizes to represent the fox, chicken and grain, and a small torn piece of paper as the boat (rowed by the farmer).

It helps to use a sheet of paper with a "river" drawn on it to simulate the situation. You an download a printout here.

Similarly, there's a puzzle called "Cannibals and Missionaries," in which three missionaries are trying to get themselves across a river filled with man-eating fish. They have to contend with three cannibals, as well, who also must get across the same river in the same canoe at the same time.

Here are the conditions:

Only two people can occupy the boat, and multiple trips can be taken in either direction. People can get  off the boat, once it gets to one side of the river or the other, and can re-enter the boat, at will. Unfortunately, if the cannibals outnumber the missionaries at any time on either side of the river, they will eat the missionaries. If this happens, the games ends unsuccessfully.

Complicating matters, only one missionary knows how to paddle the canoe, although all the cannibals know how to do so.

How to get all six people across the river in one piece?

In this case, I use three coins of one type to represent the missionaries (only one of which is heads up, indicating he can paddle the canoe), and three of another type as the cannibals.

Likewise, it helps to have the piranha-infested river drawn on a sheet of paper as a backdrop for the game, and a small piece of torn paper to use as the boat. Click here to download the backdrop.

In both games, the analogy to mathematics is found in the fact that, at each and every stage of the game, only one move makes sense and isn't obviously problematic. Patient, mindful examination of each potential move at each decision point leads one inevitably in the right direction – just as it does in solving math problems.

Can you do it? Give these a try!

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Test Your Mindfulness

How strong are your mindfulness muscles?

To find out, try your hand at these deceptive, simple-looking tasks.

In the first case, you job is to simply read the sentence contained within the triangle. As you read the sentence, write it down on a piece of paper. With luck, you'll see what's going on, here.

In the second case, try to say the COLORS of the words written, NOT the words themselves, in order, while reading at a normal pace. If you mess up, start again. Do not slow down to a snails pace in order to succeed in getting to the end of the list (although even that might not work).

Mindful, attentive concentration is a critical prerequisite for success in mathematics, and for quite a number of other activities, as well. Mindfulness is a capacity that can be trained, and just like any other training regimen, one begins with weakness but gets gradually stronger with practice and determination.

To find more puzzles for practice, search the internet.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

The Matchless Enthusiasm of Martin Gardner

The hyper-competitive struggle for survival against the old Soviet Union made American math and science education a top priority during the Cold War decades of the 20th century.

Many Baby Boomers (like me) remember with fondness the fascinating articles published monthly in Scientific American magazine. One of the most interesting features of the magazine in our time was Martin Gardner’s column on recreational mathematics, which ran for 25 years. 

Among the many necessary qualities of truly great teachers, enthusiasm might be listed first. An instructor’s genuine, overflowing enthusiasm is that which excites students' souls and convinces them that the required academic work and sacrifice will be amply rewarded. The etymology of the word “enthusiasm” (en-theos: literally, "in God") points straight at the Divine, and no one could excite the soul with the beauty of mathematics like Gardner could.

A 1998 article by the master preserves for modern readers the flavor of Gardner’s contagious enthusiasm and gold-medal exposition that so characterized his column, presenting to Gardner fans and neophytes alike the pure noetic joy that accompanies deep dives into the realm of creative mathematics.

Reflecting the timelessness of the subject, the article reads as if it were penned yesterday, fresh and new. It’s not long, and is well worth a bit of your time:

A Quarter-Century of Recreational Mathematics.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Car Talk Puzzlers

NPR's uproariously funny Car Talk could be the single funniest thing on radio. On more than one occasion it was all I could do to maintain control of my car as I listened to the hilarious rantings of Tom and Ray Magliozzi, better known as "Click and Clack ... the Tappet Brothers," during their weekly automotive question and answer show (to get an idea, check out the current list of Car Talk staff members, presumably vetted by the show's ace legal firm: "Dewey, Cheetham, and Howe").

Far from being among the dimmer bulbs in the lamp, these grease monkeys are actually scientists with MIT degrees, no less. A featured part of each broadcast is the weekly "Puzzler," and a fair number of these fabulous brain twisters are mathematical rather than automotive in nature (here's where the guys show their MIT roots).

Click here for an archive of Puzzlers from past shows ... or here to buy a recently published collection of favorite Car Talk Puzzlers at amazon.com for as little as $.01 plus S & H!

To listen to Car Talk (online, NPR affiliate station, Sirius Satellite Radio, podcasting, etc.), click here.

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Arithmetic, Algebra, Mathemagic

Arithmetic can be called the study of "known" numbers, or calculation. Algebra, then, is the study of "unknown" numbers.

Arithmetic is easy. All it takes is a good teacher, and sufficient practice.

Likewise, since all real numbers (whether known or unknown) obey the same rules, algebra is easy – provided the student is well taught and well practiced.

Knowing simple algebra empowers one to do some pretty interesting and impressive things, including all kinds of "mathemagic" tricks, like the one below (involving just basic algebra).

Give it a try!

Here goes:

1. Start with the number of doors in your home.

2. Multiply by 2.

3. Add 5.

4. Multiply by 50.

5. Add the number of legs on a normal moose.

6. Subtract 335.

7. If you’ve already had your birthday this year, add this year; otherwise, add last year.

8. subtract the number of days in July.

9. Add 85.

10. Subtract the year you were born.

11. Add 29.

12. Subtract the number of ears you have.

The three or four digit number you now have reveals the number of doors in your home followed by your age.

For more, see: Mathemagics: How to Look Like a Genius Without Really Trying (pictured above) by math wizards Arthur Benjamin and Michael Shermer.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

2 = 1

All kinds of falsehoods can be "proven" true if subtle errors in reasoning are allowed to go unnoticed. I get a kick out of debunking these faulty arguments.

See if you can find what's wrong with the well-known "proof" that 2 = 1, outlined below:

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Given.

a = b

Multiply both sides by a.
a² = ab

Add a² on both sides.
a² + a² = a² + ab

Simplify the left side.
2a² = a² + ab

Subtract 2ab from both sides.
2a² - 2ab = a² + ab - 2ab

Simplify right side.
2a² - 2ab = a² - ab

Factor 2 out of each term on left side.
2(a² - ab) = a² - ab

Divide both sides by a² - ab.
2(a² - ab)/(a² - ab) = (a² - ab)/(a² - ab)

Which "proves:"
2 = 1

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Can you find the error?

Come on ... don't look ahead until you at least give it a try!

All right, here's the mistake:

Look at the eighth step. If a = b (given), then a² - ab = 0. Division by zero is impossible, and therefore not allowed. Breaking the "never divide by zero" rule breaks the proof.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Can Your Computer Read Your Mind?

The first time you visit this site and follow the instructions, you may think your computer has come to life and has found out how to read your thoughts. You might even suppose our machines are ready to take over the world (bringing to mind “Hal,” the infamous on-boight supposeard computer in Stanley Kubrick’s masterpiece film “2001: A Space Odyssey”).

Try it a few times:

http://www.albinoblacksheep.com/flash/mind

You won’t believe it. Seems impossible. Take a minute or two right now and see if this doesn’t make you scratch your head in amazement.

But have no fear ... it’s all just simple algebra!

Here’s how it works:

Any two digit number has a ten's digit called "t" and a unit's digit called "u." The value, v, of any two digit number (and therefore of the original two digit number you pick from the puzzle page) is 10t + u. The sum of the two digits, s, is: t + u. You are instructed to subtract the digit sum from the original number; let's call this result "r."

Therefore:

r = v – s = (10t + u) – (t + u) = 9t.

Since t can only be a whole number from 0 through 9, n = 9t can only be a multiple of 9 less than 90: 0, 9, 18, 27, ... 81. So, no matter which two digit number you initially select, you will end up looking up a multiple of 9 in the chart! The writer of the program on this site has been careful to place the same symbol next to each multiple of nine in each version of the chart shown to viewers, and has instructed the program to expose that symbol in the "crystal ball" when the user clicks on it.

Go back to the puzzle page. No matter how many times you open a new and totally different puzzle page, you will notice that, on each new page, all multiples of 9 less than 90 are always associated with exactly the same symbol (the other numbers will never come up, and so they are assigned random symbols).

Here's an entertaining variation of the game:

First, send someone a number/symbol chart together with a letter the game's instructions. Then, after this person tells you he's received the letter, has calculated his "magic number," and has found and focused on their "magic symbol," you mail him another letter in which you tell him which symbol you "psychically" picked up (of course, this is the symbol associated with each multiple of 9 on the rigged chart you sent).

For even more fun:

After your victim receives the initial letter from you, you might like to place a harmless bet with the victim (say, for lunch at the winner's favorite restaurant), that you will, in fact, be able to "perceive" his magic symbol through your superior psychic powers. When you successfully "intuit" his symbol, you’ll not only get a good laugh out of it, but a free lunch with a good friend as well (unless your friend is unusually adroit with numbers, he will totally miss the pattern underlying this trick).

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Thought Boxes Revisited

Last month, I presented a puzzle that I asserted a bright first grader would find easier to solve than a bright well-educated adult.

One would think that education and experience would be advantages that should make solving any puzzle easier, not harder.

Not necessarily true.

Such is definitely not the case with the "OTTFFSSEN" puzzle, which is more easily solved with an open, uncluttered, "beginner's mind" than with a trained, sophisticated, educated mind.

One of the characteristics of human intelligence is that as it becomes more and more educated, more and more experienced, it "learns" to craft specialized shortcuts (generalized assumptions or "boxes") that increase the probability of quickly finding dependable answers to questions and viable solutions to problems.

Normally, this is a good idea. Sometimes, however, it isn't.

This is one of those times.

An adult assumes unconsciously that the "OTTFFSSEN" puzzle I presented last month must have a complicated solution, since if it didn't, it wouldn't merit consideration or attention in the first place. Unfortunately, this puzzle has a very simple solution, so looking for a complex one guarantees much needless frustration, at best. The adult's unconscious aversion to thinking "outside the box" dooms him or her to failure.

A bright young child, however, makes no such assumption of necessary complexity, and since he or she has only really studied two things thus far in school, letters (early spelling) and numbers, easily notices the pattern in the given letters:

One
Two
Three
Four
Five
Six
Seven
Eight
Nine
(Ten)
(Eleven)
(Twelve)

The answer to the puzzle, therefore, is of course:

TET.

(Click here to go to Part 1.)

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Mathematical Mind Reading Revisited

Here’s another mathemagic trick you can learn that will astound your friends, neighbors, children, parents – and instill in them a new respect for the entertainment value of basic algebra. It's not hard to learn, and might work on your math teacher, as well.

Instructions:

Pick any number, and you must remember it (the victim thinks whatever, maybe 2, or 7 – but you think "x").

Multiply by 4. (Victim thinks: 8 or 28, you think 4x.)

Add 5. (Victim: 13 or 33, you: 4x + 5.)

Add 6. (Victim: 19 or 39, you: 4x + 11.)

Subtract 3. (Victim: 16 or 36, you: 4x + 8; notice both parts of “4x + 8” divide evenly by 4.)

Divide by 4. (Victim: 4 or 9, you: 1x + 2 or simply x + 2; notice that you've now returned to "x," the original number.)

Now ... SUBTRACT YOUR ORIGINAL NUMBER. (The “x” is taken away! victim: 2 or 2, you: 2.)

You now know exactly what your victim is thinking! So, now, you can add, subtract, multiply, and divide to your heart's content, knowing with absolute certainty that you are accurately “reading” your victim’s mind. Finally, at the end of your list of instructions, announce with great fanfare (rubbing your temples, etc.) what number the victim is thinking of ... and notice with wry smile the dropped jaw and blank stare of amazement.

Note: to avoid mistakes that could spoil your performance, first hand your victim a calculator to use, and be sure that your first two instructions are to multiply and/or add; this way you'll avoid the possibility of your victims having to struggle with troublesome negative numbers.

This trick is especially effective with a group of victims (say, an entire classroom of fellow students, or the guests at a dinner party) all following your instructions simultaneously. No matter what numbers the various victims think of initially, as soon as you give the "subtract the original number" instruction, the variable is eliminated, and you are ALL now thinking of the same number, no matter what. VERY impressive! That's the beauty of algebra: that you can work with unknown numbers just as you do with known numbers, because numbers are numbers, whether known or unknown, and always obey the same rules.

With practice, entirely new routines similar to the example above can be improvised on the spot, at will, several times in a row if necessary, to convince your victim of your uncanny psychic powers. You can then, if you wish, show your victims (preferably with pencil and paper handy) how easy it is to perform the trick using basic algebra.

Suggestions:

• First practice this trick by yourself several times, using paper and pencil, playing the roles of victim and mind reader, until you're very confident that you can easily and correctly perform it.

• It’s a good idea to give your victim a calculator to work with (so they don’t make mistakes – which will make you look bad).

• Make your second instruction an addition command, to avoid having to work with negative numbers.

• It's helpful to obfuscate the trick involved by instructing victims to "add the numbers of fingers in your left hand" or "divide by the number of A's in America" instead of merely saying "add 5" or "divide by 2."

Have fun!

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

This Sentence Is False. True Or false? (Conclusion)

For convenience, I’ll repeat the sentence in question, immediately below this one.

This sentence is false.

The sentence in bold-face immediately above this one is not true, and is not false. It is “undecidable,” and has no “truth value.” The reason is rooted in its self-referential nature.

Here it is:

According to classical Aristotelian logic, what we call “common sense,” a statement cannot be both true and false. It must be one or the other.

If the sentence we’re considering is true, then it is false (after all, that’s what it directly states, that it is a false sentence). It's impossible for a sentence to be both true and false, and so we can rule out the possibility that the sentence is true.

So, the sentence in question must be false (which aside from "true," our initial assumption, is the only remaining alternative).

However, the sentence in question cannot be false either. If the sentence is false, then it is a lie, and the opposite of what it states must in fact be true. Since the sentence in question states that it is a false sentence, the opposite must in fact be the case – i.e. the sentence must be true. Again, no statement can be false AND true, so we must also throw out the possibility that the sentence in question is false.

The sentence in question is therefore demonstrably not true AND also not false – according the rules of “common sense” logic.

Weird. Apparently, logic is sometimes illogical (another strangely self-referential idea)!

If you like this kind of thing, you’ll love Hofstadter’s book (see part 1 of this investigation, here). However, be forewarned. It’s easily the "thickest” book I’ve ever attempted to read. I’m not just referring here to the length of this prodigious tome, but mainly to the difficulty of grappling with the ideas presented within it. Reading that book is like trying to step on your right foot with your right foot, nearly every page of it! It’s about radical, "outside the box" observation and expansion of the human mind – literally thinking hard about thinking hard. An intellectual challenge of the first order, this meta-logical journey will be truly maddening for most mere mortals, but is a sublimely rewarding experience for those brave and stout enough to make the trip.

For further study or additional masochism, try a google search on “self reference” or “fuzzy logic.”

Good luck!

(Click here to go to Part 1.)

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Sudoku Fever

From a March 19, 2006 article appearing in the Lincoln Journal Star:

“It seems like everyone with Sudoku fever has the same story: They stumble upon it accidentally in some book or newspaper, play it once or twice to test their skill and ride the slippery slope into Sudoku addiction.

Take Jason Brewer, for example. Flipping through the pages of a newspaper last year, he stumbled upon his first Sudoku puzzle.

Looking at the blank grid peppered with numbers here and there, he wasn't quite sure what to make of it.

But he was intrigued.”

Read the rest of the article here.

Get more info on Sudoku by visiting these sites:

http://www.sudoku.com/

http://www.soduko.org/

http://en.wikipedia.org/wiki/Sudoku

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This Sentence Is False. True Or false?

Seems like a simple question.

Uh, no. Not really.

Like a dog chasing it’s tail, brushing the bristles of the brush you’re brushing with, seeing your eyes with your own eyes (impossible, unaided), or drawing a circle around all circles on a page (try it!), the first sentence in the title of this post is strange, to say the least.

I’ll repeat the sentence in question, for convenience, immediately below this one.

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This sentence is false.

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What makes this so weird is that the sentence in question is an example of “self-reference.” Like a brush brushing itself, eyeballs seeing themselves, or circles circling themselves, the sentence in question “sentences” (i.e. refers to) itself.

[“Self reference” is the subject of a wonderful book titled: Godel, Escher, Back: An Eternal Golden Braid, by Douglas Hofstadter. I highly recommend it.]

Is the bold-faced sentence three paragraphs above this one true, or false? That’s the question.

The answer might surprise you.

(Click here to go to Part 2.)

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Mathematical Mind Reading

The old "Sum and Difference" game is a simple trick that even very young children can play to amaze and frustrate playmates or (better yet) adults (even teachers) whose algebra skills have atrophied.

It's a great way to introduce "unknown" numbers to beginning algebra students.

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Instructions:

Pick any two numbers. Add them (first number plus second number), and report the total. Then subtract them IN THE SAME ORDER (i.e. first number minus second number), and report the total.

You can determine the two numbers using simple algebra:

1st number = x (perhaps: 7).

2nd number = y (perhaps: 12).

Let: a = x + y, and let: b = x – y. The player will report to you the values of "a" and "b," the sum and difference of the two numbers (in our example, these numbers are 19 and -5). Add these results in your head (in this case, the sum is 14), and what you have will always be twice one of these numbers [since a + b = (x + y) + (x – y) = 2x.] Divide by two (14 ÷ 2 = 7), and you now have one of the numbers. Since you already know the sum of the two numbers, subtract to find the second one (19 – 7 = 12).

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An entertaining variation is the "Birth Date Trick, below.”

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Instructions:

Add your birth date plus birth month (example: for June 17, 1990, the birth date would be 17, the birth month would be 6, and the total would be 23), and report this number. Then, subtract these same two numbers IN THE SAME ORDER (i.e. birth date minus birth month). Again, be careful – this second result could be negative (in our example, the difference would be 6 – 17 = -11). Since this is really just another “sum and difference” exercise like the one above, all you have to do is add these two results and divide by two to get the birth month, then subtract the birth month from the total to get the birth date!

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Thought Boxes (OTTFFSSEN)

Sometimes, it helps to be a first grader.

The human mind is truly amazing, no doubt about it. Still, it can get in its own way, as the following puzzle illustrates.

A bright first grader would find the following puzzle easier to solve than a bright well-schooled adult, given what the first grader has probably studied so far in school.

See if you can solve it (remember ... think like a first grader!):

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Find the next three letters in this sequence:

O, T, T, F, F, S, S, E, N, ...

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(Click here to go to Part 2.)

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