Paris and Nicole’s Mayhem

In their frequent shopping sprees Paris and Nicole often have trouble checking the bill. To address their less than impressive numeracy skills they decide to return to school. After a year of maths training the teacher wants to test the class of 25 students and lines them up in a queue such that Paris and Nicole are standing next to each other. The teacher then writes a whole number on the board and the first person in the queue says “That number is divisible by 1.” Then the second person in the queue says “That number is divisible by 2,” and so on till the final student in the queue says “That number is divisible by 25.” After all this the teacher exclaims “Well done! Except for Paris and Nicole everyone made a correct statement.”  Find where Paris and Nicole were standing in the queue.


So, we’re looking for a number that can be divided by all numbers from 1 through 25 EXCEPT two of them, that are also consecutive. The number 1 will divide anything, so, that’s trivial. We can also come up with a number that can be divided by ALL of the numbers by simply taking the product of all of those numbers. This doesn’t quite give us our answer, and it also results a number that is much bigger than we really need since some factors would be repeated. (e.g. any number for which 4 is a factor will also have 2 as a factor, so if we don’t need to include both 4 (i.e. 2 x 2) and 2 as factors. Take this a bit further, you see that by including 16 as a factor (2 x 2 x 2 x 2) we include 2, 4, and 8 as well.)


Let’s create such a product, expressed in a form that is prime-factored, though assuming all of the students’ answers were correct:
2: prime 
3: prime 
4: 2 x 2 
5: prime 
6: 2 x 3 
7: prime 
8: 2 x 2 x 2 
9: 3 x 3 
10: 2 x 5 
11: prime 
12: 2 x 2 x 3 
13: prime 
14: 2 x 7 
15: 3 x 5 
16: 2 x 2 x 2 x 2 
17: prime 
18: 2 x 3 x 3 
19: prime 
20: 2 x 2 x 5 
21: 3 x 7 
22: 2 x 11 
23: prime 
24: 2 x 2 x 2 x 3 
25: 5 x 5
Combining these (e.g. the factored forms of 4 and 8 are implicit in the factorised 16) we get
2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 x 11 x 13 x 17 x 19 x 23 = 26771144400. 
This number is divisible by all of the numbers from 1 to 25. It should be easy to see that by removing, say, 13, we get a number that is still divisable by all of the other numbers from 1 through 25. (13 x 2, the smallest of the other numbers, equals 26, which falls outside range of factors.) Removing 13 we get
2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 x 11 x 17 x 19 x 23 = 2059318800,
But, we need TWO CONSECUTIVE numbers. Suppose we stick with 13 and try removing 12 as well. But, 12 = 2 x 2 x 3, so, we’d have to either remove both of the 3’s as factors, or at least 3 of the 2’s to have no 12. Suppose we remove both 3’s to give us
2 x 2 x 2 x 2 x 5 x 5 x 7 x 11 x 17 x 19 x 23
Now we’d have no 13, no 3, no 6, no 9, no 15 … already we have more than two numbers that don’t divide whatever was written on the board. So, 12 won’t work. How about 14? Same problem. You’d need to remove either all of the 2’s or the 7, but, removing all of the 2’s gets rid of all even numbers as valid factors, and removing the 7 gets rid of 14 and 21 as well. So neither Paris nor Nicole occupy the 13th place in line.

Let’s try 17. Like 13, when you remove this as a factor from the original factorization, above, you get a number that is divisible by all other numbers except for 17. We then need to look at 16 and 18 as possible second candidates. 16 = 2 x 2 x 2 x 2. Take out just one of those 2’s and we can still keep every other product in which 2, 4, or 8 is a factor, so removing just one 2 results in the loss of only 16 as a factor. Looks like Nicole and Paris were the 16th and 17th students in the line. For completeness we can look at 18 and see that it doesn’t work since 18 = 2 x 3 x 3, so we’d need to drop all of the 2’s which means we’d have a product with more than two wrong answers in the list.

So, our answer is
2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 x 11 x 19 x 23 = 787386600.
To see if I got the right answer, I googled the problem and found this site (page 3) with the exact same problem and a solution. They get the same places for Paris and Nicole as I did (16 & 17) but their number (that the teacher wrote on the board) is different: 2362159800. This looks like it’s about 3 times what my answer is, and in fact it is exactly 3 times my answer. In other words, they have an extra “3” in their factorization, for whatever reason. So, both answers are correct, but mine is the SMALLEST number that is a factor of all numbers from 1 through 25 excepting 16 and 17.

Radical Maths

This little puzzle popped up the other day and I thought it would make a good, simple example of how to deal with square roots.

First, a little terminology:  The word for root in latin is radix from which we get words like radius and radical. Mathematicians refer to the square root symbol √ as the radical symbol, and the stuff we’re taking the square root of they’ll refer to as what’s under the radical.

We start with this.

\sqrt{x + 15} + \sqrt{x} = 15

We can get rid of square root symbols by just squaring what’s under them, which will leave just the terms under the radical and nothing else.   Since this is an equation, we’ll need to do this to both sides:

(\sqrt{x+15} + \sqrt{x})^2 = 15^2 = 225

When we multiply this out, it looks like a real mess:

x + 15 + 2(\sqrt{x+15}\sqrt{x}) + x = 225

We can combine the two x’s to make this

2x + 15 + 2(\sqrt{x+15}\sqrt{x}) = 225

But, it isn’t all that much simpler.  Let’s try another approach.   Having two different radicals on the same side of the equation is what makes this messy, so, let’s move one of them to the other side of the equals sign.   We can move the \sqrt{x} term by subtracting it from both sides.

\sqrt{x + 15} = 15 - \sqrt{x}

Now let’s square both sides of the equation like we did before and see what we get:

x + 15 = 225 - (2)(15)\sqrt{x} + x = 225 - 30\sqrt{x} + x

I know, it doesn’t look all that much simpler, but, notice how there is a solitary x term on both sides?  We can get rid of that by subtracting x from both sides:

x + 15 -x = 225 - 30\sqrt{x} + x -x

leaving us with

15 = 225 - 30\sqrt{x}

It’s looking easier already!   Let’s subtract 15 from both sides, and then add 30\sqrt{x} to both sides.  We get

30\sqrt{x} = 225 - 15 = 210

Now we have simple terms for both left and right hand sides of the equation.   Divide both sides by 30 to get

\sqrt{x} = 210 / 30 = 7

Square both sides (yes, again) to get rid of the radical, and we end up with

x = 49

Let’s see if this is right.  Plug 49 in for x in the original equation:

\sqrt{49 + 15} + \sqrt{49} = 15

We can add the 49 and 15 under the first radical to get

\sqrt{64} + \sqrt{49} = 15

The square root of 64 is 8 (8 x 8 = 64) and we already have square root of 49 being 7, so

8 + 7 = 15

So, the answer checks out!

Totally RADICAL, eh?

What does the Quran really say about a muslim Woman’s Hijab? (TED Talk)

In recent times, the resurgence of the hijab along with various countries’ enforcement of it has led many to believe that Muslim women are required by their faith to wear the hijab. In this informative talk, novelist Samina Ali takes us on a journey back to Prophet Muhammad’s time to reveal what the term “hijab” really means — and it’s not the Muslim woman’s veil! So what does “hijab” actually mean, if not the veil, and how have fundamentalists conflated the term to deny women their rights? This surprising and unprecedented idea will not only challenge your assumptions about hijab but will change the way you see Muslim women.

Samina Ali is an award-winning author, activist and cultural commentator. Her debut novel, Madras on Rainy Days, won France’s prestigious Prix Premier Roman Etranger Award and was a finalist for the PEN/Hemingway Award in Fiction. Ali’s work is driven by her belief in personal narrative as a force for achieving women’s individual and political freedom and in harnessing the power of media for social transformation. She is the curator of the groundbreaking, critically acclaimed virtual exhibition, Muslima: Muslim Women’s Art & Voices.

Linear Thinking

I read this article in MotherJones a few months ago when it first came out. I thought the author’s analysis of how rural Trump voters think was particularly insightful given that many of the people the author came to know during his time among them largely agree with it. He summarized it this way:

You are patiently standing in the middle of a long line stretching toward the horizon, where the American Dream awaits. But as you wait, you see people cutting in line ahead of you. Many of these line-cutters are black—beneficiaries of affirmative action or welfare. Some are career-driven women pushing into jobs they never had before. Then you see immigrants, Mexicans, Somalis, the Syrian

Yard of Trump Supporter
Yard of Trump Supporter (photo: Stacy Krantitz)

refugees yet to come. As you wait in this unmoving line, you’re being asked to feel sorry for them all. You have a good heart. But who is deciding who you should feel compassion for? Then you see President Barack Hussein Obama waving the line-cutters forward. He’s on their side. In fact, isn’t he a line-cutter too? How did this fatherless black guy pay for Harvard? As you wait your turn, Obama is using the money in your pocket to help the line-cutters. He and his liberal backers have removed the shame from taking. The government has become an instrument for redistributing your money to the undeserving. It’s not your government anymore; it’s theirs.

What these poor souls fail to understand is that There Is No Line.

Yet, people have been sold the notion that there is a line, and it has become the basis for an otherwise unfounded sense of entitlement. “Good things come to those who wait.” How often do we hear that? Antithetically, we also hear “God helps those who help themselves.” Which is it?

Confused? Maybe that’s the point. Maybe that’s the purpose of constantly pounding conflicting messages into people’s heads: to confuse them. And if you make sure those same people never develop even a basic capacity for critical thinking, you can keep them befuddled, confused, dependent on authority figures to tell them what to do throughout their lives.

We liberals are told we don’t know how to talk to this part of the country, to these people. We’re told they feel disconnected, feel that their country has been taken over by those who they see as cutting in line, robbing them of their due. Well, to us liberals, they sound like petulant children, whining “It’s not fair!” when they feel they’ve been denied the cookie, the ice cream cone, equal turns (or time, right down to the last darn second) on the swing, or suffer any one of seemingly thousands of such injustices. The naive parent tries to reason with their child, tries to explain the how and why. Sooner or later, many of these parents (myself included) resign themselves to the reality that they are trying to reason with people who are unreasonable, and long-winded, thoughtful explanations are being ignored. The best answer turns out to be the simplest, and applies here, too: Life is not fair. Get used to that.

The overwhelming majority of people who think this way live in states that in fact receive more from the federal government than their residents pay out in federal taxes. If anything, those of us who live and work in “liberal” states, like New York or California, should be crying about how unfair that is. We don’t. Liberals operate on the principle that we’re all in this together, and that the purpose of any benevolent government is to be a tool for all of us to use to make life better for all of us. Does that mean each and every one of us will receive an equal portion? Ideally, sure; in reality, it’s not possible nor is it a reasonable expectation.

In an ideal world, life is fair, no one ever goes without, and everyone is always happy. I think the rural, benighted folk of the heartland need to grow up, need to understand that life indeed is not fair, the universe does not owe them anything, and that their life will be what they make of the opportunities chance bestows on them coupled with the character they exhibit when chance kicks them in the gut.

There is no line.

Judging for Ourselves

References to and comparisons with Hitler’s rise may be valid, however, I think a more accurate comparison can be drawn with Romania’s pre-WWII  “Iron Guard” movement  and its leader Corneliu Codreanu.   This became the basis for Eugene Ionesco’s “Rhinoceros”, a play that depicted the transition of a “normal” group of people into a herd of these animals, one by one, each with their own “reasons” for accepting them and then becoming one themselves.   We read this in high school (mid-70s) and it always stuck with me.

Right-wing fascism has reared its ugly head many times throughout history, wearing many masks.   The players, populations, and languages differ, but, the circumstances and warning signs are nearly always the same:

The good news is that, notwithstanding claims of “the death of liberalism” that fascists and their apologists seem often chant these days, it is in fact fascism, authoritarianism that is typically short-lived.   Like its economic concubine, speculative capitalism, it survives on the necessarily increasing output and consumption of it’s own excrement, and soon meets its end by starvation, or sepsis from having swallowed too much of its own shit.   Democracies, or at least regimes that recognise their power as being derived from the compliance (or complacency) of the governed survive for centuries; authoritarian regimes rarely last more than a few decades at most.

I’d be willing to bet that the articles of impeachment against Trump were drafted even before he took office — possibly even before the election.   Both Putin and the Republicans saw him as a “useful idiot” and they’ll continue to pull his strings until the FBI and other investigative agencies have all of the ducks lined up to bring him down.  At which point, Trump will either be convicted and removed, or he’ll resign, or 25A will be invoked.  In any case, Pence will be installed and, whereas Trump was a useful idiot, we’ll now have to deal with a complete moron until 2020.   With any luck, by then enough of the benighted fly-over (by then they’ll be more aptly-named fucked-over) states will have seen the folly of voting for such as Trump and use their votes to say “fuck you” to the Republicans, this time.

In the mean time, however, we’ll see a rise in anti-anyone-who-ain’t-white-christians-like-us violence and threats to persons and American Democracy itself.  The last, best defense, there, will be the courts.  My greatest fear, is that we’ll start to see judges assassinated, as has happened when such regimes take over in Latin America.   We can worry ourselves over the circus going on in Washington and state capitols across the country, but such assassinations are, to me, far more worrisome.   Two of our government’s three branches have been weakened by three decades of relentless, right-wing undermining of people and principles, leaving the judiciary to stand against it, alone.  Unless and until we see that start to happen, I have great confidence in the ability of our constitution and the democracy that it implements to endure and prevail.

Horse lookin’ at you, kid

I have been on a horse precisely three times in my life. This photo was taken on the first such occasion. About five minutes afterward, the horse, named Winnie1, was running around the paddock at what seemed like about 100 mph with me hanging off to one side, held there only by my foot, caught in the opposite stirrup. I was four or five at the time.  My aunt  (whose horse it was) and uncle (the guy in the photo NOT on the horse) quickly ran over, got the horse under control and spent the rest of the day attempting, with varying degrees of success, to calm me down — or, at least get that silent, open-mouthed scream look off of my face before my parents showed up to take me home.

I did not get back on a horse again until I was maybe 20.  A friend from school owned a horse and asked if I’d like to come along with her to feed and brush it.  She also planned to ride her a bit around the paddock. I reckoned there’d be no harm (to me) in that, so, I went along. The horse was nice and calm (turns out it was faking, plotting when and how it would make its attempt on my life) and lulled me into a false sense of trust. My friend finished her short ride, and knowing my apprehension of horses, asked if I’d like to just sit in the saddle for a bit.  Sure.   Why not.

I was still alive and well after a whole two minutes on horseback, so she (my friend) asked if I’d like to have a short walk with her (the horse) around the paddock.  “Uh … ok.”  After all, I thought, I was little when that other horse nearly killed me.   I’m a full-grown adult now.  I’m big enough to control this animal.  And if not,  I’d at least be able to get in a couple of good hits before this one did me in.

Five minutes later, there I was hanging on for dear life as that fuckin’ horse went racing around the paddock at what seemed like around 100 mph.   “Pull on the reins and say ‘Whoa!'”, my friend yelled.  I pulled and pulled on the reins, to no effect. “Whoa!!”. Nothing. “Stop!! Fermo!! Halt!! стоите!!”  The horse was evidently deaf in addition to being homicidal.

Still moving at break-neck speed,  the horse turned and headed straight for the (closed) paddock gate.  Just when I thought my fear of injury or death couldn’t be any greater, my friend opened up a whole new world of pants-shitting terror for me when I heard her hollering, “Don’t let her jump!!!!”.

I decided the only chance I had of getting off alive was to jump off while the horse was in motion.  I got my feet out of the stirrups and vaulted2 as best I could off the horse, still holding on to the horn on the saddle.  My feet no sooner hit the ground than the horse stopped dead in its tracks, maybe a meter or two from the gate. It swung it’s head around to look at me with a murderous glint in its eye that said, quite unambiguously, “Next time, asshole.”

The third (and so far, last) time was on a visit to Turkey around 14 years ago. We were on the island of Heybeliada, in the Sea of Marmara near Istanbul. There are no cars there; only horses and horse-drawn carts. My companion had never been on a horse and was dying to try it. There was a guy there with several horses for hire and he was happy to lead the horse by its tether for those who wanted to ride but were untrained. I figured, what could go wrong? So, we hired a couple of them and he led us for two or three kilometers under a perfect blue sky along one of the lovely, quiet, car-less green avenues that follow the island’s shoreline.

Our guide saw that both horses were behaving well, and my friend seemed to take to riding like a duck to water. He saw that I had taken to riding like a duck to bowling.   Still, the horse was well-behaved as it continued clip-clopping along, so he decided to let go of the tethers.  Both of them.   My friend and her horse kept right on going, the horse happily obeying her gentle tugs this way or that on the reins, the two of them making shallow S-turns as they ambled along.  My horse plodded along lazily, disinterested, slightly behind hers.   I just held the reins nice and steady, making every effort to not remind the beast that it still had me on its back.   Several blessedly uneventful minutes passed and I started to think that maybe horses didn’t have it in for me after all.

Of course, he was just biding his time, waiting until the man wasn’t looking, at which point he turned right around and headed off toward … I don’t know where … a quiet spot to murder me, perhaps.  In any case, the man soon noticed I’d been kidnapped and quickly came back for us.  He gave me an apologetic smile as he took up the tether, and continued to lead the horse, with me still on it, for the remainder of our hired time.

I’m not kidding when I say they want to kill me.

To be fair, I have had equine encounters that were not life-threatening.  One could even go as far as to say that they were pleasant.   While visiting a long-time friend in Arizona some years ago, she introduced me to her horse, Mage.   We eyed each other cautiously at first, but it wasn’t long before she was letting me pat her on the forehead and taking a carrot from my hand3.  By the end of our visit, Mage and I were getting on famously.  The trick, I found, is in knowing where you and the horse stand with one another.  I stand on one side of the paddock fence, and the horse stands on the other.  No miscommunication, and no one gets hurt.

My most recent encounter was with a neighbor’s two Australian miniatures.   These look either like fat ponies or rather well-built donkeys. Regardless, they turned out to have the sweetest most gentle temperament of any animal I’ve been close to that’s larger than, say, a great dane.4  Still, I kept to my side, they kept to theirs, and we all got along just fine and all walked away with no injuries.   I hear they’d be great for keeping what passes for a lawn in our garden under control.   I’m even tempted to see if one day we might get one.  Trouble is, these adorable little creatures live for 40 or more years.   Given my present age the odds are that, for them, it would simply be a waiting game.