## A Multiplication Based Logic Puzzle

### 440 Can You Appreciate a Good Paradox?

Brainden.com gives several examples of paradoxes as well as this definition: “A paradox is a statement that contradicts itself or a situation which seems to defy logic.”

Alan Parr suggested I write no more than the fact that 440 is the sum of consecutive prime numbers. It may seem to defy logic that he would want to try to figure out what those consecutive primes are without any hints, but he proved yesterday he is up for the challenge. Check the comments to see if he (or somebody else) can figure out what those prime numbers might be. I am waiting to see if he can again defy logic and figure them out. In the meantime, the factors of 440 will be listed at the end of this post.

I remember watching the episode of Star Trek in which several androids malfunctioned because they couldn’t handle paradoxes, but most humans seem to be able to handle paradoxes just fine. I’ve written one myself that I hope you will enjoy:

I overheard a pair of docs in the middle of a heated argument. One of them said, “As I have stated hundreds of times before, no matter what I say, YOU ALWAYS contradict me!” The other doc shook his head saying, “No, no. That’s not correct at all. YOU’RE the one who’s always contradicting ME!” I hate to hear arguing, so I left, but I have to wonder if they ever worked it out or are they still arguing about contradicting each other even today?

You might not expect to find paradoxes in the The New Testament, but crosswalkblogdotcom describes several of them beautifully in I-will-take-paradox-for-200-alec.

Shakespeare and Wordsworth used them as well. See examples in Is-this-phrase-grammatically-and-semantically-correct?

When I first joined twitter, this is part of the first conversation I had:

Steve Morris has written about paradoxes on several occasions including his very popular post titled Failure-to-fail. If we fail to fail, we become the biggest failure.

It is a paradox to have to be willing to fail in order to really succeed. As Michael Jordan stated in a Nike commercial: “I’ve missed more than 9000 shots in my career. I’ve lost almost 300 games. 26 times, I’ve been trusted to take the game winning shot and missed. I’ve failed over and over and over again in my life. And that is why I succeed.”

We will face setbacks, trials of faith, and other paradoxes in our lives, but we must not give up. All of them are opportunities for us to learn and grow. We actually ought to be grateful for them, because as we face them, we become stronger and more successful. Zeno’s paradox would have us believe that we can never reach our goals. That is simply not true. We can reach them if we keep them in our sights and keep going toward them relentlessly.

Here are the factors of 440:

• 440 is a composite number.
• Prime factorization: 440 = 2 x 2 x 2 x 5 x 11, which can be written 440 = (2^3) x 5 x 11
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 440 has exactly 16 factors.
• Factors of 440: 1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440
• Factor pairs: 440 = 1 x 440, 2 x 220, 4 x 110, 5 x 88, 8 x 55, 10 x 44, 11 x 40, or 20 x 22
• Taking the factor pair with the largest square number factor, we get √440 = (√4)(√110) = 2√110 ≈ 20.9762

Since 440 = 20 × 22, we know that 441 = 21 × 21.

1. Well, so far it’s defied 20 minutes of my logic, and I’m beginning to wonder exactly what’s up your sleeve. I am supposed to be doing other things, you know, but if no-one comes to my aid I’ll have another look tomorrow.

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2. Ok , here’s a good hint. You can use the number 17 in your answer, not once, but two times.

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• ivasallay said:

How did you know that so quickly?

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• When I realized that it had to be more than a seven number stretch it seemed logical to start adding the primes from the beginning. I had thought that when the sum reached above 440 I would begin to subtract the primes from the beginning of the list.,,,which turned out to be unnecessary. It was a delightful discovery.

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• ivasallay said:

I love how you found it! I also really like the hint you gave. It said so much without giving everything completely away!

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• it was a cruel hint. It was crafted to appear that I didn’t understand the problem. I apologize.;)

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• ivasallay said:

I think it was clever, not cruel. I am thrilled that more than one person tried to find the answer, and there was a little competition to find it first! You won fair and square.

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3. With some help from Dr Google’s calculator:

2+3+5+7+11+13+17+19+23+29+31+37+41+43+47+53+59

Somebody must have given a name to numbers which are made by summing the first n prime numbers – yes, here we are: http://mathworld.wolfram.com/PrimeSums.html

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• ivasallay said:

I could not think of a way to warn you that this even number was made up of an odd number of primes without totally giving the answer away. Bravo to you for looking for the answer honestly for 20 plus minutes!

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• I failed to take Paula’s hint, and so started from exactly the wrong direction – trying three primes, then four, until I finally hit 17. How silly of me.

Eventually I simply typed out a list of all the primes from 2 to 101, and dropped sets of them into the Google bar until I – eventually – found the solution. So it’s now become a not very interesting process at all. There’s no particular skill or insight needed, and I’m now ready to pounce if you offer 442 and 443.

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• ivasallay said:

I suppose it would have been helpful if I had told you that it would take more than 10 primes and that would not have given too much away. I think there can be a process and skill required. This one just broke the rules.

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4. abyssbrain said:

This comment is intentionally left blank.

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• ivasallay said:

I’m not sure what the mathemagician means by that.

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• abyssbrain said:

Haha, you would see something similar in some books

If the page is supposed to be blank, then why did they write something on it? 🙂

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• ivasallay said:

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• We were in a greeting card shop this afternoon. A card shows lots of cars at speed on a multi-lane motorway. At a sharp bend there is a huge sign complete with an array of flashing lights on the roadside – “Avoid unnecessary distractions while driving”!

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• ivasallay said:

What would be the occasion for sending that particular greeting card? Congratulations on getting your first driver’s license?

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5. The first 17 primes, which includes the number 17….

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• ivasallay said:

Precisely.

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6. Thank you very much for the mention, and for the paradoxes!

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• ivasallay said:

I know you enjoy paradoxes, and I have enjoyed reading the ones you’ve written. I don’t know if the mention is very helpful since I suppose almost everyone who reads my blog regularly reads yours.

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• Iva, every mention is most welcome and very much appreciated! Besides, every link to a blog will help to improve its ranking on Google.

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• ivasallay said:

Then I won’t hesitate to mention your blog again. I hope it gets a very high ranking on Google!

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7. I think the card that wbhs62 saw was probably a get well card.

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• ivasallay said:

A get-well-after-your-car-accident card?

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• Yes. Precisely.

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• Sorry to be so imaginative, but I checked today, and it’s a Happy Birthday card.

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• ivasallay said:

For a sixteen year old or for those of us who are speeding to old age faster than we’d like?

Are you going to answer the challenge to express 442 as the sum of consecutive primes?

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8. Great posts worth seeing from ivasallay:

How to Solve a FIND THE FACTORS Puzzle
A Forest of 240 Factor Trees
440 Can You Appreciate a Good Paradox?

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