A Multiplication Based Logic Puzzle

Archive for June, 2017

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!

  • 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:

 

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816 and Level 2

Eight is half of sixteen, so 816 is divisible by 6. You probably weren’t expecting that divisibility rule, but it’s true.

816 can also be easily divided by 2, 4, and 8. How many factors does 816 have in all? Plenty! Scroll down past the puzzle and see!

  • 816 is a composite number.
  • Prime factorization: 816 = 2 x 2 x 2 x 2 x 3 x 17, which can be written 816 = (2^4) x 3 x 17
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 816 has exactly 20 factors.
  • Factors of 816: 1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816
  • Factor pairs: 816 = 1 x 816, 2 x 408, 3 x 272, 4 x 204, 6 x 136, 8 x 102, 12 x 68, 16 x 51, 17 x 48 or 24 x 34
  • Taking the factor pair with the largest square number factor, we get √816 = (√16)(√51) = 4√51 ≈ 28.5657

Since 17 is one of its factors, 816 is the hypotenuse of a Pythagorean triple:

  • 384-720-816 which is 48 times 8-15-17

816 is repdigit OO in base 33 (O is 24 base 10). That is true because

  • 24(33¹) + 24(33º) = 24(33¹ + 33º) = 24(33 + 1) = 24 × 34 = 816

816 is the sum of the sixteen prime numbers from 19 to 83:

  • 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 816

Coincidentally, 816 is also the sixteenth tetrahedral number.

That’s because 16(16 + 1)(16 + 2)/6 = 816, which is a fast way to compute it. Here’s what it means to be the 16th tetrahedral number:

815 and Level 1

Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

  • 489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

 

  • 815 is a composite number.
  • Prime factorization: 815 = 5 x 163
  • 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 815 has exactly 4 factors.
  • Factors of 815: 1, 5, 163, 815
  • Factor pairs: 815 = 1 x 815 or 5 x 163
  • 815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

814 and Level 6

814 is even, so it is divisible by 2. What else is it divisible by? Let’s apply 11’s divisibility rule:

  • 8 – 1 + 4 = 11, so 814 can also be evenly divided by 11. (If the sum of every other digit minus the sum of the missed digits is divisible by 11, then the number is divisible by 11.)

This Level 6 Puzzle is a doozie, but I assure you that it CAN BE be solved using logic without guessing and checking!

Print the puzzles or type the solution on this excel file: 10-factors 807-814

Now here’s a little bit more about the number 814:

  • 814 is a composite number.
  • Prime factorization: 814 = 2 x 11 x 37
  • 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 814 has exactly 8 factors.
  • Factors of 814: 1, 2, 11, 22, 37, 74, 407, 814
  • Factor pairs: 814 = 1 x 814, 2 x 407, 11 x 74, or 22 x 37
  • 814 has no square factors that allow its square root to be simplified. √814 ≈ 28.530685

Because 37 is one of its factors, 814 is the hypotenuse of a Pythagorean triple:

  • 264-770-814 which is 22 times (12-35-37)

814 is also repddigit MM in BASE 36 (M is 22 base 10) because 22(36) + 22(1) = 22(36 + 1) = 22 × 37 = 814

813 My Newest Grandchild

My NEWEST grandchild was adopted a couple of months ago in China. My daughter-in-law blogged about picking up their daughter and returning to her orphanage to say good-bye before they left China. The details given are very moving. In spite of the traumatic start, this little girl and her family have grown to love each other very much.

Here she is sitting with my husband, me, and her big sister. My husband and I are pretty new to her so she’s probably thinking in Mandarin, “Who are these people?” On the other hand, we are delighted to be a part of her life now.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors 807-814

Here’s a little about the number 813:

  • 813 is a composite number.
  • Prime factorization: 813 = 3 x 271
  • 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 813 has exactly 4 factors.
  • Factors of 813: 1, 3, 271, 813
  • Factor pairs: 813 = 1 x 813 or 3 x 271
  • 813 has no square factors that allow its square root to be simplified. √813 ≈ 51315.

813 is repdigit 111 in BASE 28 because 28² + 28¹ + 28º = 813

Stetson.edu and my calculator informed me that 813^e = 81,366,615.06223032 . . .

 

812 How Many Triangles in All?

Here’s a puzzle for you. How many total triangles are there in the figure below?

Okay, if you guessed 812 because this is my 812th post, you would be right. But what if you were asked that question in some real world situation where accessing the internet to get the answer isn’t permitted. How would you know the answer then?

True, you might have memorized the formula I mentioned in 658-How Many Triangles Point Up? How Many Triangles Point Down? How Many Triangles in All?:

  • The total number of triangles = ⌊n(n+2)(2n+1)/8⌋ where the brackets mean round decimals DOWN to the closest integer. 
  • Here n = 14, so the number of triangles is 14×16×29/8 = 812. Rounding down wasn’t necessary since the product of two consecutive even numbers is always divisible by 8.

Still, you probably wouldn’t remember that formula unless you had seen it VERY recently or you have a photographic memory.

You could actually COUNT all the triangles. In the post about 658 total triangles, I noted that the 13 rows of small triangles formed a total of 169 of the smallest triangles, but I suggested that it would be easier to ignore that nice square number and instead count the number of triangles pointing UP separately from the number pointing DOWN. You will add up a lot of triangular numbers as you sum up the number of them pointing up and again as you sum up the number pointing down.

Making a chart of the number of triangles pointing UP would be easy. It’s just a list of triangular numbers in order. However, the chart for the ones pointing down might be confusing because you don’t use all of the triangular numbers, and the ones you use will be different for an even number of rows than for an odd number of rows. For example,

  • the pointing DOWN portion of the chart for 14 rows of triangles below uses these seven triangular numbers: 1, 6, 15, 28, 45, 66, and 91,
  • while the pointing DOWN chart for 13 rows of triangles uses six different triangular numbers: 3, 10, 21, 36, 55, and 78.

Here is a chart listing the number of triangles of any size that are contained in a triangular figure made with 14 rows of small triangles. Interesting note: Because 14 + 2 = 16, a multiple of 8, the total number of triangles in this case will be divisible by 14, the number of line segments on each side.

Making such a chart works. However, remembering  which triangular numbers to use and how many you should use, especially when counting the odd number of rows of triangles pointing DOWN, might be difficult.

Today I was thinking about triangular numbers and their relationship to square numbers:

  • The sum of two consecutive triangular numbers always makes a square number. On the chart above, I’ve paired up certain consecutive triangular numbers by coloring them the same color. The sizes of the triangles being paired together by like colors are not the same size (with one exception), but their sums are nevertheless square numbers every time.
  • Thus, we also can come up with a single list of numbers to add to arrive at the total number of triangles as illustrated below. This has made me changed my mind about using those square numbers to help count! Notice how every other number on each list below, including the last number to be added, is a square number.

Of the methods discussed in this post, this one that includes square numbers might be the easiest one to remember. Here are the steps I used to find the total number of triangles:

  • Count the rows and determine if that number is even or odd.
  • Write the triangular numbers in order until the amount of numbers written equals the number of rows.
  • If the number of rows is even, replace the 2nd, 4th, 6th, etc. entries with corresponding square numbers: 4, 16, 36, etc.
  • If the number of rows is odd, replace the 1st, 3rd, 5th, etc entries with corresponding square numbers: 1, 9, 25, etc.
  • Add all the numbers remaining in the list. The sum will be the total number of triangles for that many rows of small triangles.

Okay, that puzzle was rather difficult and took a while to explain. You may find this Level 4 puzzle to be easier:

Print the puzzles or type the solution on this excel file: 10-factors 807-814

Here’s a little more about the number 812:

  • 812 is a composite number.
  • Prime factorization: 812 = 2 x 2 x 7 x 29, which can be written 812 = (2^2) x 7 x 29
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 812 has exactly 12 factors.
  • Factors of 812: 1, 2, 4, 7, 14, 28, 29, 58, 116, 203, 406, 812
  • Factor pairs: 812 = 1 x 812, 2 x 406, 4 x 203, 7 x 116, 14 x 58, or 28 x 29
  • Taking the factor pair with the largest square number factor, we get √812 = (√4)(√203) = 2√203 ≈ 28.4956 (That was three multiples of 7)

812 = 28 × 29, which means it is the sum of the first 28 even numbers.

  • Thus, 2 + 4 + 6 + 8 + . . .  + 52 + 54 + 56 = 812

It also means that we are halfway between 28² and 29², or halfway between 784 and 841. The average of those two numbers is 812.5.

AND it means that 28² + 29² – 1 = 2(812) = 2(28 × 29)

Since 29 is one of its factors, 812 is also the hypotenuse of a Pythagorean triple:

  • 560-588-812 which is 28 times 20-21-29.

811 My Youngest Grandchild

My youngest grandchild is 7 months old and is just beginning to crawl. I love the way he wipes drool off his own face in this video. It’s pretty funny.

 

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors 807-814

  • 811 is a prime number.
  • Prime factorization: 811 is prime and cannot be factored.
  • The exponent of prime number 811 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 811 has exactly 2 factors.
  • Factors of 811: 1, 811
  • Factor pairs: 811 = 1 x 811
  • 811 has no square factors that allow its square root to be simplified. √811 ≈ 28.4780617

How do we know that 811 is a prime number? If 811 were not a prime number, then it would be divisible by at least one prime number less than or equal to √811 ≈ 28.5. Since 811 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 811 is a prime number.

811 is the second half of a twin prime, and it is also the sum of five consecutive prime numbers:

  • 151 + 157 + 163 + 167 + 173 = 811.

Stetson.edu informs us that the smallest prime factor of 24!+1 is 811. Wow! Really?

  • If you type 24!+1 into your computer calculator you get 620,448,401,733,239,439,360,001.
  • If you divide that number by 811, you get 765,041,185,860,961,084,291.
  • If you type that into Number Empire’s Prime Number Checker, you will see that it is prime.

  • Thus 811 is the smaller of 24!+1’s two prime factors!

 

 

 

 

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