# Tag Archives: brainteaser

## Mathematical fanatical 2019 – the answers.

Hi folks!

In yesterday’s post, you’ll recall how I gave you, dear readers mathematicians, a mathematical brainteaser: how to get ‘2019’ using the four main arithmetic operations [+, -, ×, ÷], plus parentheses [(, )’], and the figures 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1; and not in any order but in the order given [10 down to 1] – and with no joining up of the numbers to make bigger numbers.

Budding mathematicians among the readers of my blog in Russian sent in their answers. Here are some of the more elegant runners-up among them:

( 10 * 9 * 8 – 7 * 6 – 5 ) * ( 4 – 3 + 2 * 1 ) = 2019 (by Skarbovoy);

( 10 + 9 ) * ( 8 + 7 + 6 ) * 5 + 4 * ( 3 + 2 + 1 ) = 2019 (by eve_nts);

(10 + 9 * 8 * 7 – 6 – 5 ) * 4 + 3 * 2 + 1 = 2019 (also by eve_nts).

A big thanks to everyone, btw, who sent in answers! I had great fun picking out the right ones, checking for mistakes – and coming up with some formulations of my own.

But back to the podium…

## Mathematical fanatical: How to turn ‘1’ into ‘2019’?!

Hi folks!

2019. Is it two thousand nineteen, or twenty nineteen? How does it sound to you? “It’s just a number – the number of the year,” you say? Hmm. I think your saying that might mean poetry or just rhythm-and-rhyme aren’t your fortes. Right? For – and not a lot of people know this – there’s a phenomenon known as digital poetry. There are all different kinds of digital poetry, as a quick glance at that Wikipedia page will show you. One kind I find rather intriguing is the one where numbers are substituted for the words of the works of the great poets – and you need to work out who that poet is based on the way the numbers are pronounced – which words are stressed, the number of syllables, and so on.

Here, for example, are some numbers that, when spoken, reflect the poetry style of Alexander Pushkin (at least, in Russian:):

17 30 48
140 10 01
126 138
140 3 501

I wonder, does it work the other way round? I mean, can a poem be made up about 2019? Might there be some budding digital poets among you, dear readers, who might be able to conjure up a poem about 2019?

Meanwhile, for the mathematicians and physicists among you, my traditional annual arithmetic puzzler…

I hope you remember the rules. If not:

You need to get the number of the current year – this year being 2019 of course – using the four main arithmetic operations [+, -, *, ÷], plus parentheses [(, )’], and the figures 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1; and not in any order but in the order given [10 down to 1] – and with no joining up of the numbers to make bigger numbers (e.g., 1 and 2 making 12).

For example:

((10 + 9 – 8) * 7) + (6 + 5) * (4 – 3 + 2) + 1 = 111

Hmm. That gives 111. But I wanted 2019.

All righty. Try it! Who does it first is the winner!

10 9 8 7 6 5 4 3 2 1 = 2019
…

Did you do it? Well done!

Now, let’s continue the fun by upping the difficulty a bit: let’s drop the 10:

## Japan quakes. Japan shrugs.

Not long after leaving Japan, I read in the Russian press that there’d just been an earthquake there causing several deaths and a ‘transportation collapse’! Oh my Geiger, I thought, and quickly looked for more details on other sites on the web. Well, sadly they were right about the deaths – a few dozen, but ‘transportation collapse’? The earthquake was registered as a 6 on the moment magnitude scale. Sure, it gives everything a real good shake – but it doesn’t knock you, the dog, or the furniture over.

And ‘trains grinding to a halt’ (the article went on)? Of course they did; they’re meant to: special systems are installed on all the railroads to make the trains do just that! And besides, in Japan, there’s a magical 15-20-second warning sent out to all cellphones before an earthquake hits! How on earth that is possible I have no idea, but it sure is massively helpful. I’ve seen it for myself (back in 2011): we were in a car and a local’s mobile emitted a warning signal (so we quickly pulled up), and 15 seconds later the lampposts and traffic-lights started shaking along the road (it turned out it was aftershocks of the (9-magnitude) 2011 earthquake).

In Japan, all buildings, all roads, all bridges, all towers, all infrastructure – it’s all designed and built specially so as to withstand strong earthquakes. Even a 9-magnitude quake damages very little at all! So magnitude-6? You can work that one out yourselves ).

Sure, there’s rail disruption. Sure, the airports aren’t firing on all cylinders. But that’s it. And after a while – everything automatically starts to move and fly again. Japan is quite ready for earthquakes; it has to be.

So, anyway… What were we doing in Japan in the first place? A few things; one of them – attending Interop in Chiba (near Tokyo):

(Required)

## The mystery of the Aldabra giant tortoise.

I first set eyes on these incredible creatures last year in 2017. Just a year later and I was back for more, and since then I haven’t been able to stop wondering: where did they come from and how did they manage to survive? Wikipedia gave me part of the answers, which in this instance I trust completely: the Aldabra giant tortoise is endemic to the island of the Aldabra atoll in Seychelles.

// Endemic, btw, refers to species – not individuals. So, let’s say you live and don’t ever venture far from a single address/location, you aren’t endemic to that place; you’re just lazy – or a hikikomori, if in Japan! :)

But how? This question keeps me up at night…

## Digital 2018 – pt. 2

Hi folks!

Quite a bit of motivation is needed to solve interesting brainteasers. Thankfully I’ve never had any trouble mustering motivation. But more about that in a bit…

First up, as per the requests of many, two brainteasers that don’t require a calculator or computer – it’s quicker using a trusty old pencil and pad. All righty…

Brainteaser No. 1

There exists a really beautiful 10-digit number. The first (left-most) digit in it is the overall quantity of 0s in this number. The second digit – the quantity of 1s. The third – 2s. And so on. The last digit is the quantity of 9s. What is that digit?

It’s not as hard as it may at first seem. To solve it you need merely (i) a head, (ii) a brain inside it, and (iii) the ability to use it. So good luck!

The second riddle is a little more difficult. Even if you have a head, a brain and ability, not everyone will get it. This one’s solving is probably reserved for arithmetic geniuses – the sort that are able to multiply large numbers in their heads. Let’s see…

Brainteaserdestroyer No. 2

Does there exist a natural (whole, nonzero, positive) number that gives upon multiplication by 2018 a result that consists of a number made up of 10 1s and/or 0s? (everyone’s a programmer here: it’s all about the 0s and 1s:). In other words, is it possible to multiply 2018 by something whole and positive so that the result of the multiplication only has 0s and 1s in it – and is 10 digits long? If yes – let’s see it! If there are many – which is the smallest, and by how much? If there are none, explain the reason why.

Ok all you smart alecks, and Alexandras, thinking caps on! For the best/funniest answers – prizes!

And now a bit on how last week’s riddle was solved:

Digital 2018 – pt. 1

How to get 2018 out of the sequence 10-9-8-7-6-5-4-3-2-1 and its truncations: 9-…-1, 8-…-1 and so on?

Here’s how:

## Digital 2018 – pt. 1.

Boys and girls!

December’s here again already. Over the next few weeks there’ll be the usual Christmassy-New-Year good vibes, then there’ll be the presents, fireworks, champagne, mistletoe, more champagne, and then the clock will strike midnight and we’ll have a +1 to the eternal yearly calendar. Then, for perhaps the first few weeks of January we’ll all still be saying and writing the date as day/Jan/the year 2017; oops, 2018! We all do it! I think ).

Twenty-eighteen. It has a ring to it; yes – a nice, round number. And each numeral that makes up the date is an even number… What? You’re not sure about 1? Come on! 1 is 2 to the power of zero. Kinda :). But wait! There’s more even-ness in this number: each digit of 2018 is a power of two. But what don’t you like about zero? Well, think of an artificial number, raising it to the power of which two gives zero – what, difficult? Now think of an imaginary ‘i’, the square of which gives -1. Come on: such a sexy number as 2018 is just crying out for working a sweat up about :).

Ok, ok; agreed. We won’t spoil arithmetic with all kinds of unnecessary chimeras, to the power of which each decent two turns into an empty zero. But then, as per Chinese tradition, eight means wealth! So get ready – 2018 should be blessed with prosperity; there’s no chance of avoiding it!

Sooo. It’s time to stretch and warm up for what is bound to be an infinitely interesting – and perfectly prosperous – year. So let’s get stuck into some 2018-related arithmetic. And what comes to mind first? Yes: evenness.

2018 = 2*1009

1009 is a prime number. A bit like 2017. Last year I promised that 2017 would be a simple, straightforward year. And look how in the end it turned out! Now we need to get ready for an extra-simple/straightforward year, aka – a minus plus a minus gives a plus.

What else? The sum of all the numerals in 2018 is 11: a most photogenic number from any angle, and one that’s dear to me for technical reasons: the product of all nonzero numerals = 16, which can’t not raise the spirits of any programmer on the planet.

Ok, enough. Warm-up over. Let’s move onto our already traditional New Year arithmetic exercise. Here we go…

Given figures: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1. Using only ‘+’, ‘-‘, ‘*’ and ‘/’, plus ‘(…)’, all in any quantity, and also using exclusively these figures only once and only in that order… – how do you get 2018?

For example:

((10 + 9 – 8) * 7) + (6 + 5) * (4 – 3 + 2) + 1 = 111

Here we get 111. But we need to get 2018!

Marks, get set, go! Who’ll do it first to become the champ?

10 9 8 7 6 5 4 3 2 1 = 2018

Once you solve it, you go to level two: Get 2018 from the same figures minus the 10.

9 8 7 6 5 4 3 2 1 = 2018

Got it? > Level 3…:

8 7 6 5 4 3 2 1 = 2018

I managed these without a calculator – and without peeking at last year’s brainteaser – in around 20 minutes in Shanghai waiting for my flight to Moscow. My attempt at the next one was interrupted of course by the inevitable ‘turn off your devices’ nonsense on the plane, but once the ‘seatbelts fastened’ light went off, I carried on where I’d left off:

7 6 5 4 3 2 1 = 2018

This one is impossible without a factorial. I think we could allow here powers and roots too.

6 5 4 3 2 1 = 2018

Here I needed a multifactorial.

All righty. From ten figures to six: done. We’re half way there. Next up will be the second part of the brainteaser: from five and down. But we’ll save that for next time. For now, I’ve a party to get to!…

Cheers!

## A Chinese gastronomic enigma.

Many of you may have noticed that I rarely write about food. Photos of food or meals on Instagram are not my strong suit :) However, it would be wrong to say I’m indifferent to food. Absolutely not! These are my favorite kinds of cuisine:

1. Chinese cuisine. To be more precise, all types of Chinese cuisine, and above all, South Xianggang cuisine (is that the proper name for it?).
2. Japanese cuisine. To be more precise, all types of Japanese cuisine with their fresh, fried, grilled, roasted, boiled, etc. food. (Which reminds me of this video about the mysteries of Japanese cuisine.)
3. All other Asian food.
4. The entire culinary spectrum of the Caucasus. (The challenge here is to stay within the confines of lunch and dinner rather than succumbing to all-out gluttony…which I don’t think is right.)
5. Borscht.
6. That’s probably enough, or we may descend into the aforementioned gluttony :)

So now, I need the help from the audience.

There is a remarkable vegetable that grows in China (or, more correctly, on Hainan island). When cooked, it looks like this:

Its name in Mandarin is 四角豆.

“Four-cornered beans” according to my translation tool. Indeed, this veg has a very distinct four-cornered stalk. When preparing it, they chop the stalk at an angle (which results in rectangles with sharp corners) and pour on some seasoning.

I’ve never seen this vegetarian dish anywhere outside Hainan, and that includes Hong Kong which is just next door. This vegetable only grows in Hainan, and that’s where it all seems to get eaten.

So, two questions.

1. What’s the proper name for this vegetable in Russian and English?
2. Just in case I’m wrong, does anyone know if this tasty veg is on sale anywhere outside China? Would be great to know.

// After all that I have a strong urge to go and have lunch :)

## 2017: Leonardo, Fibonacci and Fermat Numbers: It’s Not So Complicated.

In my previous post we had a math competition. Let me remind you of the task:

Using +, -, x, ÷ and (), make the row of numbers from 10 to 1 equal 2017.

That was an easy task, which got more complicated.

How about 9 to 1 with arithmetic equalling 2017? 8 to 1? 7 to 1? And down to just 1?

Before I could say ‘What January blues when you’ve got arithmetic in your life!?!’ I had answers from people in our fan club streaming in! And some of them (remember, there are different possibilities to get the same answer of 2017) were so wonderfully interesting, while others were so interestingly not-quite-elegant enough, that, well, I just had to share some of them with you…

## 2017: Prime Numbers, Factorials, Primorials, Derangements: It’s Complicated.

As many will already know, the number 2017 is a prime number; that is, it can be divided without a remainder only by itself and 1. Must say, the theory of prime numbers is a wholly interesting one and an extremely useful one too, as any cryptographer will tell you :).

But today I’ll be writing about something different. See, based on the fact that 2017 is prime – or ‘simple’ – many, myself included, are anticipating a simple, straightforward and calm year 2017, especially since 2016 was a bit of a rotter. Let me show you why.

Like I said, prime numbers are those that can only be divided by themselves and 1 without leaving a remainder. Non-prime numbers are called composite numbers, incidentally.

Turns out that 2016 is not only a composite number but a very composite number! It has a whole eight divisors. Grab a calculator your smartphone and test it for yourself:
2016 = 2 * 2 * 2 * 2 * 2 * 3 * 3 * 7

Whoah! Even the quantity of divisors is anything but simple, since 8 = 2 * 2 * 2.

So what about other years? Was 1917, the year of the Russian Revolution, a ‘prime’ year, for example? No, it wasn’t. 1917 = 3 * 3 * 3 * 71. Just four divisors, but they’re kinda poignant – and prophetic of nothing much good.

So what about other very prime/simple years, and other very non-prime/non-simple ones? Ok, let’s narrow this down a bit to 1980 through present day…

Prime/simple years:
1987
1993
1997
1999
2003
2011

And in the near future there are a few more prime/simples:
2027
2029

(eek, that’s a lot of non-simple years until then)

The most non-prime/non-simple years were:
1984 = 2 * 2 * 2 * 2 * 2 * 2 * 31 (seven divisors)
2000 = 2 * 2 * 2 * 2 * 5 * 5 * 5   (also seven)

There were six divisors in 1980, and there’ll be six in 2025. All other years can be called semi-prime/semi-simple.

But I digress…

Now, in the popular British mathematical journal The Guardian :), readers were recently teased with a… brain teaser. In the blanks between the sequence of figures 10 9 8 7 6 5 4 3 2 1 you need to add arithmetic symbols (+, -, x, ÷, (),) – as many as you like – so as to get the number (year) 2017.

For example, if you add arithmetic signs as follows you get 817:
10 * 9 * (8 + 7 – 6) * (5 – 4) + 3 * 2 + 1 = 817

But how do you add arithmetic to get 2017?
10?9?8?7?6?5?4?3?2?1 = 2017

Come on, have a go!

As for me, in nine minutes I got the equation to equal 2017 by kinda wonky arithmetic (I made the ‘3’ and ‘2’ = ’32’!); then, in around 15 or 20 minutes I got the answer in a proper way without bending the rules. I say ‘a’ way: there are different ways of getting to 2017!

So, tried it yet?

Ok, let’s make it harder: Now take away the 10:
9  8  7  6  5  4  3  2  1 = 2017

## Happy New Year from Central Moscow!

Happy New Year folks, and hope you all had great holidays!

You won’t believe this… but this post is about… RED SQUARE! // Incidentally, the square I consider to be the most beautiful spot in Europe!

I hadn’t been in downtown Moscow on New Year’s Eve since… oooh, 15 years ago! Yep – 2001 was the last time, on Pushkin Square watching the fireworks. But I’d never been on Red Square on New Year’s Eve. What?! So this year I decided to make amends…

So how was it? Well, actually, my overall impressions were… mixed. And it’s those mixed impressions that I’d like to share with you today.