An Eulerian Graph is a graph whereby all the vertices within the graph have an even degree. This definition is obtained from Euler’s Theorem which was published in 1736.

**Theorem (Euler 1736): **A connected graph is Eulerian if and only if every vertex has an even degree.

Using this theorem, it is easy to prove that House and House X Graphs do not have an Eulerian Path. An Eulerian Path is a path whereby each edge is visited exactly once.

Using GraphTea, I’ve constructed a House Graph and a House X Graph. I’ve labeled each vertex within the graph with it’s corresponding degree. You can easily see that both graphs do not possess the law of Euler’s Theorem, and therefore these graphs are not Eulerian.

A Hamiltonian Graph is quite similar to an Eulerian Graph, however, the conditions are different. A Hamiltonian Graph is where there exists a cycle which can visit each vertex within the graph exactly once, with the exclusion of the start and end of the cycle.

I attempt to show if the House Graphs are Hamiltonian, or at least, if they contain a Hamiltonian path.

Using Ore’s Theorem (1960), we can investigate if the House X Graph and the House Graph contain a Hamiltonian Cycle.

**Theorem (Ore 1960): **Let G be a finite and simple graph on n 3 vertices. If deg v + deg u n for every pair of non-adjacent v and u vertices of G, then G is Hamiltonian.

I’ve created a line between each non-adjacent pair of vertices within both graphs. We can see that the House Graph is non-Hamiltonian and the House X graph is Hamiltonian.

Graduate Texts in Mathematics: Graph Theory (Diestel)

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It’s a nice summary, covering the origin of the theorem, what it means, how it’s used and how probabilities calculated from that theorem differ from conventionally calculated probabilities. It’s a lot of information packed into a readable and concise format. Here’s one bit about the Monty Hall problem. (Try telling one of your friends about it and then tell them that they should always switch doors. Odds are they won’t believe you!):

A famously counterintuitive puzzle that lends itself to a Bayesian approach is the Monty Hall problem, in which Mr. Hall, longtime host of the game show “Let’s Make a Deal,” hides a car behind one of three doors and a goat behind each of the other two. The contestant picks Door No. 1, but before opening it, Mr. Hall opens Door No. 2 to reveal a goat. Should the contestant stick with No. 1 or switch to No. 3, or does it matter?

A Bayesian calculation would start with one-third odds that any given door hides the car, then update that knowledge with the new data: Door No. 2 had a goat. The odds that the contestant guessed right — that the car is behind No. 1 — remain one in three. Thus, the odds that she guessed wrong are two in three. And if she guessed wrong, the car must be behind Door No. 3. So she should indeed switch.

Jasen Rosenhouse wrote a whole book about the Monty Hall problem, which you can find at this link.

Anyway, Faye manages to write the whole article without once showing the theorem itself, but nevertheless captures its essence. It’s good popular science writing. And here’s the theorem:

Addendum: Faye also has a new piece in *Forbes* which will interest many of us: it’s about why plants make caffeine. The piece is called,”Though it may taste divine, coffee DNA tells a Darwinian tale.” I won’t give you the punchline, but here’s a teaser:

]]>But caffeine appears to be extra beneficial to plants since it evolved through different genetic mutations in coffee, chocolate and tea – a phenomenon called convergent evolution. We know why we like coffee, but what’s in it for these different plants?

The basic formula is:

part/whole = %/100

You can use this to find out the percentage of something, for instance what percent is 14 out of 52?

14/52 = %/100

Multiply both sides by 100 to get % by itself:

(100)14/52 = (100)%/100 : 1400/52 = %

Now divide 52, which leaves us with: 26.9%.

You can also use it to find out what a percent equals in real terms. For example, what is 12% of 74?

x/74 = 12/100

Again, multiply both sides so the variable is by itself, in this case 74:

(74) x/74 = (74) 12/100 : x = 888/100 : x = 8.88

Note: If you are trying to calculate the percentage of change, it is the same basic formula except on the bottom is your starting point and on top is the change in real terms. Percent and 100 are the same as on the other side of the equation. If you are calculating change, this can be a negative number if the change is going in a negative direction.

Ex: You start with 10 units and end up with 15 units. What is the % of change?

Start is easy – 10 units. Change is the difference between end and start: 15-10=5.

5/10 = %/100

(100) 5/10 = (100) %/

500/10 = %

50 = % The change was 50%.

Note: Another thing to watch for: if you are asked to find out two percents, for example, if you loose 50% and then gain 50%, the answer is **not** the number you began with. For example, 50% of 100 is 50: 100 – 50 = 50. However, 50% of 50 is 25: 50 + 25 = 75. Yep, you would need 100% for 50 to get back up to 100.

Life’s much simpler when we take things one at a time, killing time softly- just like when we do basic arithmetic. Things were easier to understand then. We just add, subtract, multiply and divide. Basic! We thought- no scratch that. I thought I’ve got the whole world in my hands.

And then came Boolean Algebra.

I was fine with the simple binary conversions and arithmetic. We were friends. ‘*Why do we always have to make things complicated?’* That’s the first thought that occurred to me when I seeped into the Boolean world.

George Boole (1815-1864) sought to give symbolic form to Aristotle’s system of logic, writing a treatise on the subject in 1854, entitled ‘An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities’, which codified several rules of relationship between mathematical quantities limited to one of two possible values: true or false, 1 or 0. Hence, his mathematical system became known as Boolean Algebra.

After a ‘lil introduction about the topic, we started to “simplify” Boolean expressions- which of course is trickier.

Deciding what axioms, laws, and theorems to use is a confusing stage for me. But oh well. YOLO na this. #jeje

UGH, Math. In hopes of finding the correct answer, just remember the words of mathematician Stan Gudder, * “The essence of Mathematics is not to make things complicated, but to make complicated things simple.” *And maybe, that would cheer you up. Somehow. Yay.

**“Chronology”**

From precognitive dreams, where the future is recognizably predicted before it happens, we can infer that time is *other* than what ordinarily we think it is. From the way philosophers have sometimes talked, mathematicians and physicists too on occasion, and theologians fairly often, we can gather that there is a dimension of reality where the hands of the clock don’t measure anything.

*That dimension is called “eternity.”*

Bracketing out eternity, the poet says,

*… at my back I always hear*

*Time’s wingèd chariot hurrying near.*

What is he trying to say to “his coy mistress”? Would it be something along the lines of

*Hurry it up, girl;*

*you’re not getting any younger;*

*what’re yuh saving it for?*

Let’s face it, eternity’s lookin’ better all the time.

Pursuing this for a moment, we think of developments like the synchronization of clocks in time zones, supposedly a side-product of the scheduling needs of cross-continental railroads. Or we may be aware of “Taylorism,” at one time a program that tried to get factory workers to synchronize like the machines with which they worked. It’s since been discredited, but was supposed to maximize efficiency. These features of the modern world have been associated with loss of personal power – here the power to set one’s own tempo.

On counts like these, chronology gets a bad rap.

Nevertheless, this is unfair to chronology. The arts of living are closely bound up with the arts of making time one’s friend. For example:

- Take the case where we suddenly feel disoriented, out of joint and faintly despairing, without knowing why.

The remedy? We retrace in memory what happened *before* we felt this way and what happened *next*, when the anomic feeling began. Suddenly we understand and the feeling lifts! We have re-secured our place in time.

- Take another case. One person accuses another of having treated her unjustly. The accused makes the counter-claim that she is the one who was mistreated.

The remedy? We try to find out who started it. The one who first began the abuses is the party at fault. The other one, who reacted in self-defense, is the victim, the injured party. If we claim indifference to these particulars, we are indifferent to justice. Is the chronology difficult to establish incontrovertibly?

*Yeah. Welcome to earth.*

When I taught Introduction to Philosophy courses, I would begin by setting down, on the longest blackboard, the history of the wider culture in which philosophy took shape. Here is the classical period with its dates and a brief characterization. Here the Hellenistic, with the same. Here is Rome and the dates and a bit more. The medieval period, the Renaissance, what is called the “Modern” period (from Copernicus to Newton), the Enlightenment, the Romantic period, the nineteenth and 20^{th} centuries and what they each were like.

Philosophy is the longest unbroken conversation in the history of the world. You can’t join a conversation without knowing, at least in a general way, what was said before you entered it.

Interestingly, my observation was that students who had attended the opening class immediately acquired a fundament of intelligence. Students who came on board late had to scramble for it. They were the ones who thought that Socrates and George Washington were contemporaries. That’s not just a gap in one’s store of information. It’s more like — *not smart*.

We live and learn in narrative form. There are some false narratives and also ones closer to the truth. I guess only God knows it all. But unless we try to attach our memories to the standard of chronological accuracy, we lose grip on the storied reality of our lives and thus on our most telling adventures.

]]>Read More at http://ift.tt/1rB1Pkz

]]>Precision almost seems like part of the definition of math. The realm of numbers and symbols and methods seems to be one in which vagueness is an outcast.

http://motherboard.vice.com/read/how-uncertain-math-might-be-better-math

]]>I fell headfirst from the pages of my linear algebra textbook into another classroom. It reminded me of calculus, but was of no building I’ve ever stepped foot in: the walls were white and discolored at the edges, darker greys and burnt yellows that made the corners stretch into oblivion. Low white tables sat in clusters of four or five around the room, but I was the only student held between its four walls. And hanging at its front, two large projector screens hung, covered in a PowerPoint slide as simple as text and a link.

But I said I dreamed of fantasy, and here the portal lay.

“This is how we know calculus has existed for thousands of years!” was the sparse caption on the slide, and reaching my hand out–for I was not the one at the computer, itself hidden from my sight–I pressed my hand to the screen, the letters appearing on the back of my hand, and it gave a bit beneath my fingers, as screens are likely to do. And then the world gave a bit, edges breaking apart and twisting into abysmal passages as new colors rushed in, a collection of dots and pixels that slowly realigned itself into another world.

I stood on a bridge. A rather large bridge, it stretched the size of a basketball court from side to side, and end to end ran at least three or four times that length. It was also a rather old bridge, built of wooden planks now weathered into a delightful cascade of brown, every inch worn down beneath the oppression of thousands, if not millions of marching feet. The rails were sparse, of the same old, greying wood, only waist high, and I knew without looking that it stood on thin wooden poles that stretched deep beneath me to a land richer than I could ever imagine–and too far away to not feel terror if I should steal a glance.

At the closer end of the bridge was what I had come to see. The bridge fanned out into a valley as verdant as the word defines, a dozen shades of green grasses, bushes, and low trees filling it up in perfect symmetry as it wrapped around a chasm, framed by a mountain in the background that rose high with beards of mist stirred slowly by its ancient breathing.

In this chasm there was a scant wooden frame, and there, three great stone carvings. At their top ends they were all the same: great, light grey stone balloons, as if they had been filled with helium and hung attached to the trellis only to keep from blowing away. But underneath there was something more: a tail grew down from each balloon’s bottom, the stonework seamless as each string twisted around and through the others, falling ever toward those unnamable depths I had mentioned before. The curvature was exquisite, the geometry non-Euclidean. In this tangle of stone as ancient, perhaps, as humanity–if not more ancient than that–I saw path integrals and the derivatives of composed functions, there were conic sections and complex mappings, there was wonder and amazement and longing to understand how such a sculpture had come to exist.

And then I tired of trying to understand and I turned away. I felt like I must be in some island nation, given the intensity of the sun and the humidity in the air, and so I began venturing across that great bridge upon which I had found myself.

I came at last to–to what? Are there human words enough to describe this place? The colors were vermillion and violet, gold and magenta and fuchsia, orange and absolute crystalline clarity that has all the playing of light but none if the ethereality, only that impermeable jewel-like quality.

That is what this world was: a land composed of jewels, rich and lustrous, scintillating beneath a sun held above an enchanting haze that turned the sky into a cloudy morass of opalescent grey and white and even purer white. The dredges of sunlight seeped through this mass, however, and all the world sparkled in colors too boggling to capture entirely: there was ruby and topaz, quartz as bright as day and as pink as roses, there were diamonds and tourmaline and sunstone.

But if all that had been all, it would have been wondrous but of no great consequence. The ground–shining with the smooth glow of all these jewels fastened into each other, forged from each other–twisted and coiled around itself like the fronds of an intricate coral, and here and there, where the ground appeared–at a distance–to dip down into a pool or an ocean, there was a greater glistening, as though the light had decided suddenly to slow down, to hang suspended in the air, tired from racing across the galaxy and ready to relax and linger once and for all. And I was not certain if these supplicating lights were water or the sunlight glaring off the jewel-encrusted ground I wandered upon.

The path I followed curved, though level all the same it remained, and I heard sudden squawking–the first blemishes of sound upon this brilliant plane. And there, sitting along the solid, softly flowing light, there were three birds: the third in line flew off as I stumbled to a stop, and the other two turned back and fluttered their wings in warning. These were birds of no sort seen on earth: they had the bills of flamingos, so black they shined like amethyst, the short legs of ducks, and the feathers of peacocks in show–but their colors were as rich and sanguine as the ground they perched upon, and I wondered briefly before I backed away if their feathers were at all feathers or simply tricks of this gemstone light…

J.R.R. Tolkien, in the essay “On Fairy-Stories,” tells us the cornerstone of fantasy is sub-creation: the enactment of a Secondary World. But fantasy is neither travel tale nor dream, as has been this exploration… Another critic of the fantastic, Farah Mendlesohn, classifies four types of fantasy, the first of which is portal fantasy–in which we enter the world of the fantastic, and the portal of dreams is an allowable entrance.

She speaks more so of style and form than does Tolkien: his intent is more rigorous, more personal. What do we gain from Fantasy? Why do we read it? What is its value among the classics, both passed down from antiquity and penned today?

He paints us a picture of Fantasy: that first plunge into a Secondary World, that broadening–by force or wit–of the imagination. He sings of Recovery: the moment when seeing the world anew through fantasy brings us back to better appreciate the world which we’ve been given in our daily lives. Next comes Escape, freeing us from the binds in which we willingly intertwine ourselves, and finally is Consolation: the sudden joyous turn that sets the world right in the end, the fulfillment of destiny, the oft-awaited Happy Ending.

This dream delivered me unto fantasy, but there the fable stops. This Cyclopean sculpture of calculus from the dawn of time could not recover my struggling love for mathematics. This enchanting, prismatic landscape could not recompile my code, stripping away stray bits of memory-stopping data until every experience is as fresh and *new* as this hyperbolic shoreline. No dream from which we wake is ever a lasting escape, and what consolation is there when we return to a world no different than before?

No, I suppose this was not fantasy. It was only a dream.

]]>http://mathsframe.co.uk/en/resources/resource/274/Reading_Numbers_Crystal_Crash *(choose level 3)*

http://www.sheppardsoftware.com/mathgames/earlymath/fruitShootNumbersWords.htm *(choose 1 to 1000)*

http://www.dositey.com/2008/addsub/NC2/numbersmatch2.htm

http://www.mathsisfun.com/numbers/ordering-game.php * (choose Ascending 1 to 1,000)*

http://www.bbc.co.uk/bitesize/ks1/maths/number_ordering/play/popup.shtml

http://www.ictgames.com/caterpillar_slider.html *(set the MAX to the maximum)*

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Everything is getting better in the Washington, DC public school system since the City Council did away with the elected school board and instituted mayoral control over the schools. Chancellors Michelle Rhee and Kaya Henderson have overseen tremendous improvement ever since, because the teacher evaluation system known as IMPACT and the removal of seniority starting in school-year 2009-10 were game-changers that have ensured continual test score increases.

After looking at the record, I beg to differ.

What the record actually shows is that with all those changes, almost none of the promises Chancellor Rhee made actually came true. Plus, if you look at how the various subgroups (blacks, whites, hispanics, ELLs, SPEDs, and so on) did, you will see almost no progress since 2009.

I will show you the results, and I think you will agree that mayoral control of the schools and the current focus on tests, tests, and more tests has not even come close to accomplishing any of the promises that were made or that citizens should expect.

Let’s first look at the promises made for elementary and secondary math and reading scores on the DC-CAS, and compare those promises with the actual results. First, elementary reading:

Remember, the big changes in the operations of DC Public Schools began in the 2009-2010 school year. Before those big changes were implemented, there had been some modes and steady improvement in the elementary reading scores on the DC-CAS, which Rhee, Kamras, and Henderson promised would continue through 2013 — and that you can see as the dotted blue line in the graph shown above. In 2014, under the law known as “No Child Left Behind Act”, every single student in every single subgroup in every single public school was supposed to be proficient, which is why my dotted line suddenly veers sharply up into dreamland at 100%. So, as you can see, the actual percentage of elementary students at the ‘advanced’ or ‘proficient’ level in DCPS in 2013 is slightly BELOW what it was in 2009, the last year before IMPACT (and the year I retired).

Next, let’s look at secondary reading:

Once again, the promised goals far outstripped the actual achievements after 2009 in secondary reading, which showed several years of small declines after 2009, and a couple of years of small increases. An astute reader will notice that for 2008 and 2009, the dotted blue line (promises) is below the solid red line. That’s because when I added up the numbers of students in grades 7, 8, and 10 who were ‘proficient’ or ‘advanced’ in those years from the official spreadsheets, I got slightly different results from what Rhee & company claimed when they made the agreements with the four large foundations. I don’t quite know what causes the difference, so I’m calling their numbers for those years the “claimed” results, and my results the “actual”. Such hubris on my part, I know…

In any case, nobody could claim that there has been steady growth in secondary reading scores in DCPS since 2009, the last year before IMPACT. Recall that 2007 was the first year that DC students took a new exam called the DC-CAS, instead of the previous test called the SAT-9. In every school district that I or other researchers have examined, when a new standardized test is instituted, it is very common for students’ scores to plummet the first year. After that, teachers and students learn how to take the test and instruction changes, and scores begin to rise again. We see that pattern here for years 2007, 2008, and 2009. But after that, frankly, the scores are very close to “flat”.

Next — elementary math scores:

Once again, we do steady increases from 2007 through 2009, which I attribute to teachers learning how to teach to a new test and students figuring it out as well. After 2009, when Rhee instituted IMPACT and made all those promises to those large foundations and the public, the growth pretty much stopped, and the gap between those promises (the dotted blue line) and the actual results (the red line) got wider. In 2014, the last year we have data for, the elementary math scores actually dropped again, not by much — but this was the year that under No Child Left Behind, 100% of all students were supposed to be proficient.

Next — secondary math:

This fourth graph is almost an exact duplicate of the pattern established with the previous three graphs. Once again, there are different results in different official documents, but the gap between the promised results and the actual results is getting wider, and there has been rather little growth since 2009.

Now let’s look at the various subgroups: African-American students, Hispanics, whites, those learning English as a foreign language for the first time, those in special education, and those eligible for free or reduced-price lunches. First, math:

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It is our birthday this week. Jay and I officially opened our tutoring centre on September 28th 2009. So we are **5** years old already. Thank you to all students who have attended over the last 5 years and helped us create the superb service we now offer. We have changed so much since those early days. However there has always been one constant; that is to give fantastic service and help your child succeed and grow in confidence.

Since our initial opening we have become well known across the island and have helped new students develop and improve in English, maths and many other subjects up to and including SAT and A level. We are very proud of our achievements, but without our students of course it would have been impossible!

My, how we have grown from just Jay and I with 5 students on day 1. We now have a wonderful team around us and over 175 student places each week.

Thank you to Martyn, Rosheen, Judith and Marga who tutor to the highest standards. To Caroline who manages our diary and the general running of the office very well. Finally we cannot forget Angie our cleaner who makes the office look ‘spick and span’ after the students have left. **Thank you to you all.**

Much of our success in growing our business has been down to clients referring our services to new people. We would like to thank everyone who has helped us previously. If you are happy with our service please do not hesitate to tell at least one more person about us!

In order to officially thank you for helping us ‘spread the word’ we offer a free lesson for each new sign up.

During the autumn term it is so important to lay solid foundations for the school year. Our mission is to support the hard work students are now undertaking.

Development, improvement, confidence or maintenance of skills let us know what you need and we will work with you and your child over the coming academic year.

**Remember together we can make a difference.**

The first NEW section is for GCSE Mathematics Past Papers. This section includes PDF’s of all papers on syllabus A (Higher and Foundation) from the June 2012 examination season onwards, accompanied with relevant markschemes and grade boundaries.

]]>“Now, a new study proves that people who are good at reading are also quite naturally talented at math.”

Actually, not really. The study, in and of itself, is quite interesting, but this blogger, apparently relying on an *LA Times* piece rather than the original paper, gets a number of the details wrong.

You can read the original paper here. The abstract reads:

“Dissecting how genetic and environmental influences impact on learning is helpful for maximizing numeracy and literacy. Here we show, using twin and genome-wide analysis, that there is a substantial genetic component to children’s ability in reading and mathematics, and estimate that around one half of the observed correlation in these traits is due to shared genetic effects (so-called Generalist Genes). Thus, our results highlight the potential role of the learning environment in contributing to differences in a child’s cognitive abilities at age twelve.”

One of the big confusions that accompanies a study like this is the misunderstanding of the concept of heritability. Heritability is *not* a measure of the percentage of a trait is caused by genes. It is not even an individual level measure of a trait, it is a measure of how much of the variance in a population is genetic.

One of the consequences of this is that heritablity is not fixed, it will be affected by the level of environmental variance. We can see this by a simple thought experiment. Imagine a society with total absolute equality between individuals. In that circumstance, since there would be no difference in environment, heritablity would be one hundred percent. This conclusion, strikes many as counterintuitive but if you contemplate it for a while you will see that this must be true.

But honestly, even on twitter, I find stuff that I never thought I’d find. There are actually some pretty funny things that I found before that were math related, and I never thought math could make me laugh. And if you ever find something that’s math related and funny, show your math teacher, believe me, they will appreciate it, and even if you don’t t think it’s funny, they will think that it is hilarious. Weather it be a math pun, or a math equation that makes a funny answer, trust me math teachers love it, and who knows you might too.

Just because it’s math doesn’t mean it can’t be funny, or mean it can’t make your math teacher laugh. Save it and show it to people, it may make someone laugh.

]]>| 65 | 64 | 63 | 62 | 61 | 60 | 59 | 58 | 57 | | 66 | 37 | 36 | 35 | 34 | 33 | 32 | 31 | 56 | | 67 | 38 | 17 | 16 | 15 | 14 | 13 | 30 | 55 | | 68 | 39 | 18 | 05 | 04 | 03 | 12 | 29 | 54 | | 69 | 40 | 19 | 06 |01| 02 | 11 | 28 | 53 | | 70 | 41 | 20 | 07 | 08 | 09 | 10 | 27 | 52 | | 71 | 42 | 21 | 22 | 23 | 24 | 25 | 26 | 51 | | 72 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

Take the number 21, which is three places across and up from the bottom left corner of the spiral. The route to the origin contains the numbers 21, 22, 23, 8 and 1, because first you move right two places, then up two places. And 21 is what I call a route number, because 21 = 3 + 4 + 5 + 8 + 1 = digitsum(21) + digitsum(22) + digitsum(23) + digitsum(8) + digitsum(1). Beside the trivial case of 1, there are two more route numbers in the spiral:

58 = 13 + 14 + 6 + 7 + 7 + 6 + 4 + 1 = digitsum(58) + digitsum(59) + digitsum(60) + digitsum(61) + digitsum(34) + digitsum(15) + digitsum(4) + digitsum(1).

74 = 11 + 12 + 13 + 14 + 10 + 5 + 8 + 1 = digitsum(74) + digitsum(75) + digitsum(76) + digitsum(77) + digitsum(46) + digitsum(23) + digitsum(8) + digitsum(1).

Then I wondered about other possible routes to the origin. Think of the origin as one corner of a rectangle and the number being tested as the diagonal corner. Suppose that you always move away from the starting corner, that is, you always move up or right (or up and left, and so on, depending on where the corners lie). In a *x* by *y* rectangle, how many routes are there between the diagonal corners under those conditions?

It’s an interesting question, but first I’ve looked at the simpler case of an *n* by *n* square. You can encode each route as a binary number, with 0 representing a vertical move and 1 representing a horizontal move. The problem then becomes equivalent to finding the number of distinct ways you can arrange equal numbers of 1s and 0s. If you use this method, you’ll discover that there are two routes across the 2×2 square, corresponding to the binary numbers 01 and 10:

Across the 3×3 square, there are six routes, corresponding to the binary numbers 0011, 0101, 0110, 1001, 1010 and 1100:

Across the 4×4 square, there are twenty routes:

Across the 5×5 square, there are 70 routes:

Across the 6×6 and 7×7 squares, there are 252 and 924 routes:

After that, the routes quickly increase in number. This is the list for n = 1 to 14:

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600… (see A000984 at the Online Encyclopedia of Integer Sequences)

After that you can vary the conditions. What if you can move not just vertically and horizontally, but diagonally, i.e. vertically and horizontally at the same time? Now you can encode the route with a ternary number, or number in base 3, with 0 representing a vertical move, 1 a horizontal move and 2 a diagonal move. As before, there is one route across a 1×1 square, but there are three across a 2×2, corresponding to the ternary numbers 01, 2 and 10:

There are 13 routes across a 3×3 square, corresponding to the ternary numbers 0011, 201, 021, 22, 0101, 210, 1001, 120, 012, 102, 0110, 1010, 1100:

And what about cubes, hypercubes and higher?

]]>I have added a small portion to my Teach 5 groups this year which I think has made even more of an impact. Before my students go to there teaching group they get to meet with the other students in the class which teach their same problem. This gives them an opportunity to make sure their problem is correct and see multiple ways to solve their problem some times.

If you have no idea what Teach 5 is check out How to set up Teach 5 and Teach 5 packet.

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