- 747 is a composite number.
- Prime factorization: 747 = 3 x 3 x 83, which can be written 747 = (3^2) x 83
- The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 747 has exactly 6 factors.
- Factors of 747: 1, 3, 9, 83, 249, 747
- Factor pairs: 747 = 1 x 747, 3 x 249, or 9 x 83
- Taking the factor pair with the largest square number factor, we get √747 = (√9)(√83) = 3√83 ≈ 27.3313007.

It’s Steve Morris’s birthday so I thought I’d make him a cake, but no regular size cake will do. He has been one of my earliest supporters, and I know that sometimes even a level 6 puzzle is just too easy for him. Once he sent out this tweet:

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```This maths puzzle took me 6 minutes – can you beat that? http://t.co/8bX7yjFKRt via @IvaSallay

— Steve Morris (@SteveMorris999) June 1, 2014

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Admittedly that puzzle was easier than most level 6’s, but recently he tweeted me a puzzle that I have had to start over more than once and still haven’t conquered:

@findthefactors @NerdintheBrain @abyssbrain Can you solve GCHQ’s Christmas card puzzle? (GCHQ is Britain’s NSA) https://t.co/YrKXBeFRMq

— Steve Morris (@SteveMorris999) December 10, 2015

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I think Steve Morris is due for an extra difficult Find the Factors puzzle for his birthday, one that all the numbers from 1 to 16 can be the factors. I’ll wait at least a week before I give any hints to complete it, too. As always, there is only one solution, but it can be found using logic.

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

The possible factors for each clue is given below. Adding 14, 15, and 16 as possible factors really complicates the puzzle!

—————————————

Now I’ll write a little about the number 747:

747 is a palindrome in base 10. Boeing’s most recognizable airplane also bares that number.

747 can be written as the sum of consecutive numbers five different ways:

- 373 + 374 = 747; that’s 2 consecutive numbers
- 248 + 249 + 250 = 747; that’s 3 consecutive numbers
- 122 + 123 + 124 + 125 + 126 + 127 = 747; that’s 6 consecutive numbers
- 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 = 747; that’s 9 consecutive numbers
- 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 = 747; that’s 18 consecutive numbers

747 is the sum of three squares six different ways. Three of those ways repeat squares.

- 27² +
**3**² +**3**² = 747 - 25² + 11² + 1² = 747
- 23² + 13² + 7² = 747
- 21² + 15² + 9² = 747
**19**² +**19**² + 5² = 747**17**² +**17**² + 13² = 747

—————————————

Since today’s puzzle is the biggest puzzle I have ever published, it could be a little difficult just noticing that a factor had been duplicated in the top row or first column. Here is the tweet Steve Morris sent out once he finally solved the puzzle:

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```@findthefactors pic.twitter.com/VwFKb50GoW

— Steve Morris (@SteveMorris999) January 24, 2016

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Now after waiting over a week, I now reveal one of the ways to solve this difficult puzzle logically:

Comments on:"747 Happy Birthday, Steve Morris!" (21)Steve Morrissaid:Wow! Iva, I am truly touched and honoured by your post! I’m amazed you even remembered it’s my birthday. What a lovely gift – a fiendishly hard puzzle to solve 🙂 I will work on it this evening and let you know if I can solve it.

That GCHQ puzzle took me 3 attempts over 3 days, but I did eventually work it out. Unfortunately it then leads to an even harder set of puzzles that defeated me.

I’m very proud to be a long-time supporter of Findthefactors, as I think it’s such a cool idea and a great way to practise multiplication facts. I’m glad your blog is getting such well-deserved attention.

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abyssbrainsaid:Happy Birthday!

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nerdinthebrainsaid:Happy, happy birthday to Steve! What a wonderful way to celebrate someone’s birthday! You’re both awesome, and I’m ever so happy to have you as blog friends. 😀

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Happy birthday to me! | Blog Blogger Bloggestsaid:[…] was also astonished to find that fellow blogger, Iva Sallay, has created a maths puzzle as a birthday gift for me. I’m very much looking forward to trying to solve that […]

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Steve Morrissaid:Well this is a really tough challenge. No solution yet …

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ivasallaysaid:I won’t have time today, but tomorrow I’ll see how long it takes me to solve it. I designed this puzzle over a month ago so I don’t remember exactly what the logic is to solve it. I promise I won’t peek at my notes of that logic. I predict it will take me at least a couple of hours.

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Steve Morrissaid:OK, I officially cannot solve this puzzle!

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ivasallaysaid:Let me give you the first piece of logic to solve it. The possible common factors for

80and64are 8 and 16, butthe common factor has to be16for the following rather complicated reason:The 32 in the puzzle must be 4 x 8 or 2 x 16 so it will use either an 8 or a 2.

22 definitely uses a 2. If 32 and 22 use both 2’s, then 8 = 8 x 1. Otherwise 32 is 4 x 8.

Therefore, either 32 uses an 8, or 8 uses an 8.

Either way, an 8 is used so 64 would have to be 4 x 16, not 8 x 8.

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ivasallaysaid:As promised, I set the timer to see how long it would take me to solve the puzzle after not looking at it for a month. I didn’t set the timer until after I gave you the first piece of logic so I remembered how to start the puzzle. Also there still were a few things I remembered even after a full month since creating it, but it still took me 26 minutes, 21 seconds to solve the puzzle today. It is a bear.

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Joseph Nebussaid:Well, a happy birthday to him! Neat.

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Steve Morrissaid:OK, yes I got that. Then several more entries fall into place and I deduced that the common factor of 48 and 72 must be 8.

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ivasallaysaid:It’s not 8. Try 6 or 12 instead.

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Steve Morrissaid:So if the common factor of 80 and 64 is 16, then 65 must be 5 across and 13 down; then 39 is 13 across and 3 down.

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ivasallaysaid:Yes.

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Steve Morrissaid:So 8 is 1×8 because 22 uses a 2, 32 uses either a 2 or (4 and 8), and one 5 has already been placed.

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Steve Morrissaid:Then I worked out that 8 must be 1×8.

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Steve Morrissaid:Next, I deduce that 84 is 6×14, 45 is 3×15 and the top 48 is 4×12. This means that the common factor of 48 and 72 must be 8!!!!

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ivasallaysaid:Figure out clue 15 before you figure out clue 8.

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Steve Morrissaid:I give up 😦

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ivasallaysaid:No-o-o-o! This puzzle was supposed to make you happy!

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Steve Morrissaid:Got it! Thanks to Iva for showing me where I was going wrong!

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