## A Multiplication Based Logic Puzzle

### 817 Tiny Squares

Start at the top row of this level three puzzle and work down one cell at a time until you’ve written each number from 1 through 12 in both the top row and the first column. You will have solved the puzzle if all the clues given in the puzzle are the products of the numbers you wrote. You can do this!

Print the puzzles or type the solution on this excel file: 12 factors 815-820

• 817 is a composite number.
• Prime factorization: 817 = 19 x 43
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 817 has exactly 4 factors.
• Factors of 817: 1, 19, 43, 817
• Factor pairs: 817 = 1 x 817 or 19 x 43
• 817 has no square factors that allow its square root to be simplified. √817 ≈ 28.58321

817 is the sum of three consecutive prime numbers:

• 269 + 271 + 277 = 817

Because it has two factor pairs in which the numbers in the pair are either both odd or both even, 817 can be written as the difference of two squares two different ways:

• 817 = 31² – 12², which I’ve illustrated below.
• 817 = 409² – 408²

Being able to be written as the difference of two squares means that 817 is a leg in two Pythagorean triples so:

• 744² + 817² = 1105²
• 817² + 333744² = 333745²

Last, but certainly not least, 817 is the difference of two consecutive cubes, namely 17³ – 16³ = 817. That means that 817 is the 17th centered hexagonal number as well! There are 17 squares on each side of the figure below and 817 tiny purple squares in all. The horizontal row in the middle has 2(17) – 1 = 33 tiny purple squares.

2(17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32) + 33 = 817.

That difference of two cubes also means that

• 817 = 17³ – 16³ = (17 – 16)(17² + 16·17 + 16²) = (1)(289 + 272 + 256) = 817.

That’s from a³ – b³ = (a – b)(a² + ab + b²) and is illustrated below using 2D cubes instead of squares. In 3D, when the sides are folded up, the darker cubes can be claimed by two different sides so the figure will look like a 17 x 17 x 17 inside corner such as in a room where three perpendicular lines meet, but in 2D it’s just the flat net you see here:

### 797 and Mathematical Ways to Love

Mathwithbaddrawings.com has some thoughtful and entertaining Ways to Tell a Mathematician that you love them.

Artful Maths wrote a post that includes beautiful mathematical origami valentines and a “string art” cardioid that is made with a pencil instead of string.

This puzzle could be another mathematical way to show some love:

Print the puzzles or type the solution on this excel file: 12-factors-795-799

Here are a few facts about the number 797:

797 is a palindrome in three bases:

• 797 BASE 10 because 7(100) + 9(10) + 7(1) = 797
• 565 BASE 12 because 5(144) + 6(12) + 5(1) = 797
• 494 BASE 13 because 4(169) + 9(13) + 4(1) = 797

But there’s one more palindromic fact about the number 797: It is the sum of two square numbers that are also palindromes!

• 797 = 121 + 676. Note that 11² = 121 and 26² = 676.

Since it is the sum of two squares, 797 will also be the hypotenuse a Pythagorean triple:

• 555 – 572 – 797 calculated from 26² – 11², 2(26)(11), 26² + 11².

797 is the sum of three squares seven different ways:

• 28² + 3² + 2² = 797
• 27² + 8² + 2² = 797
• 24² + 14² + 5² = 797
• 24² + 11² + 10² = 797
• 22² + 13² + 12² = 797
• 21² + 16² + 10² = 797
• 20² + 19² + 6² = 797

797 is also the sum of the 15 prime numbers from 23 to 83:

• 23 + 29 + 31 + 37+ 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 797

Here is the most basic information about the number 797:

• 797 is a prime number.
• Prime factorization: 797 is prime and cannot be factored.
• The exponent of prime number 797 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 797 has exactly 2 factors.
• Factors of 797: 1, 797
• Factor pairs: 797 = 1 x 797
• 797 has no square factors that allow its square root to be simplified. √797 ≈ 28.231188.

How do we know that 797 is a prime number? If 797 were not a prime number, then it would be divisible by at least one prime number less than or equal to √797 ≈ 28.2. Since 797 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 797 is a prime number.

Here’s another way we know that 797 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 11² = 797 with 26 and 11 having no common prime factors, 797 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √797 ≈ 28.2. Since 797 is not divisible by 5, 13, or 17, we know that 797 is a prime number.

### 790 and Level 3

• 790 is a composite number.
• Prime factorization: 790 = 2 x 5 x 79
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 790 has exactly 8 factors.
• Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
• Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
• 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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Here’s a little more about the number 790:

There are four ways to make 790 using three squares:

• 21² + 18² + 5² = 790
• 27² + 6² + 5² = 790
• 23² + 15² + 6² = 790
• 22² + 15² + 9² = 790

790 is the hypotenuse of Pythagorean triple 474-632-790 which is 3-4-5 times 158.

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### 784 and Level 3

If you know the multiplication facts up to 12 x 12, then it is obvious that 784 is divisible by 7. However, 784 is divisible by a whole lot more numbers than that. 784 can actually be evenly divided by 15 different numbers. Hmm, 15, that’s an odd number. A number’s factors always come in pairs. One of its factor pairs must contain the same factor twice, meaning 784 is a perfect square!

• 784 is a composite number.
• Prime factorization: 784 = 2 x 2 x 2 x 2 x 7 x 7, which can be written 784 = (2^4) x (7^2)
• The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 784 has exactly 15 factors.
• Factors of 784: 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784
• Factor pairs: 784 = 1 x 784, 2 x 392, 4 x 196, 7 x 112, 8 x 98, 14 x 56, 16 x 49, or 28 x 28
• 784 is a perfect square. √784 = 28

But that’s not the only thing remarkable about this perfect square: √784 is 28, the 7th triangular number, so like all other squared triangular numbers 784 has this additional property:

Just as 784 is a perfect square, five of the twelve clues in today’s puzzle are also perfect squares. But don’t let that fact trick you into writing the same factor in both the first column and the top row every time!

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s more about the number 784:

784 can be written as the sum of three squares in exactly one way:

• 24² + 12² + 8² = 784

784 is also a palindrome in Bases 13, 17, and 27:

• 484 BASE 13; note that 4(169) + 8(13) + 4(1) = 784
• 2C2 BASE 17 (C is 12 base 10); note that 2(289) + 12(17) + 2(1) = 784
• 121 BASE 27; note that 1(27²) + 2(27) + 1(1) = 784

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### 776 and Level 3

• 776 is a composite number.
• Prime factorization: 776 = 2 x 2 x 2 x 97, which can be written 776 = (2^3) x 97
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 776 has exactly 8 factors.
• Factors of 776: 1, 2, 4, 8, 97, 194, 388, 776
• Factor pairs: 776 = 1 x 776, 2 x 388, 4 x 194, or 8 x 97
• Taking the factor pair with the largest square number factor, we get √776 = (√4)(√194) = 2√194 ≈ 27.85677655

Try solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Here’s more about the number 776:

776 is the sum of two squares: 26² + 10² =776.

776 is the hypotenuse of Pythagorean triple 520-576-776 so 520² + 576² = 776².

776 is also the sum of three squares five different ways:

• 26² + 8² + 6² = 776
• 24² + 14² + 2² = 776
• 24² + 10² + 10² = 776
• 22² + 16² + 6² = 776
• 18² + 16² + 14² = 776

776 is a palindrome in three other bases:

• 646 BASE 11; note that 6(121) + 4(11) + 6(1) = 776
• 272 BASE 18; note that 2(18²) + 7(18) + 2(1) = 776
• 161 BASE 25; note that 1(25²) + 6(25) + 1(1) = 776

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### 770 and Level 3

• 770 is a composite number.
• Prime factorization: 770 = 2 x 5 x 7 x 11
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 770 has exactly 16 factors.
• Factors of 770: 1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770
• Factor pairs: 770 = 1 x 770, 2 x 385, 5 x 154, 7 x 110, 10 x 77, 11 x 70, 14 x 55, or 22 x 35
• 770 has no square factors that allow its square root to be simplified. √770 ≈ 27.74887.

Here is a puzzle for you to solve:

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Because 5 is one of its factors, 770 is the hypotenuse of a Pythagorean triple, and 462² + 616² = 770².

770 is the sum of the squares of three consecutive numbers: 15² + 16² + 17² = 770.

770 can also be written as the sum of three squares seven other ways:

• 27² + 5² + 4² = 770
• 25² + 12² + 1² = 770
• 25² + 9² + 8² = 770
• 24² + 13² + 5² = 770
• 23² + 15² + 4² = 770
• 20² + 19² + 3² = 770
• 20² + 17² + 9² = 770

770 is palindrome MM in Base 34 (M = 22 base 10); note that 22(34) + 22(1) = 770.

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### 763 and Level 3

It’s obvious that 763 is divisible by 7 so it is a composite number.

• 763 is a composite number.
• Prime factorization: 763 = 7 x 109
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 763 has exactly 4 factors.
• Factors of 763: 1, 7, 109, 763
• Factor pairs: 763 = 1 x 763 or 7 x 109
• 763 has no square factors that allow its square root to be simplified. √763 ≈ 27.6224546.

Now try solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s a little more about the number 763:

763 is the sum of consecutive numbers three different ways:

• 381 + 382 = 763; that’s 2 consecutive numbers.
• 106 + 107 + 108 + 109 + 110 + 111 + 112 = 763; that’s 7 consecutive numbers.
• 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 = 763; that’s 14 consecutive numbers.

763 is also the sum of consecutive prime numbers two different ways:

• 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 = 763; that’s 9 consecutive primes.
• 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 763; that’s 17 consecutive primes.

763 is the hypotenuse of a Pythagorean triple, and 420² + 637² = 763².

763 is also the sum of three squares two different ways:

• 27² + 5² + 3² = 763
• 23² + 15² + 3² = 763

763^4 = 338,920,744,561, a number in which every digit appears at least one time. Stetson.edu informs us 763 is the smallest number whose 4th power can make that claim.

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