# 806 and a Level 6 Dunce Cap?

When I put this post together I took a second look at today’s puzzle and thought, “That looks a little like a dunce cap.” That thought led me to two very interesting articles whose information surprised me greatly.

The first one titled “The Dunce Cap Wasn’t Always so Stupid” explains that long ago when the dunce cap was first introduced by the brilliant Scotsman John Duns Scotus, it became a symbol of exceptional intellect. In fact wizard hats were most likely modeled after them. Unfortunately, this positive perception of the caps remained for only a couple of centuries.

The second article is short but helped me visualize Topology’s Dunce Hat. I enjoyed watching the animation of this mathematical concept.

I hope you will enjoy trying to solve the puzzle. It is a Level 6, so it won’t be easy. If you succeed, you’ll deserve to feel that you have exceptional intellect.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Now here is some information about the number 806:

• 806 is a composite number.
• Prime factorization: 806 = 2 x 13 x 31
• 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 806 has exactly 8 factors.
• Factors of 806: 1, 2, 13, 26, 31, 62, 403, 806
• Factor pairs: 806 = 1 x 806, 2 x 403, 13 x 62, or 26 x 31
• 806 has no square factors that allow its square root to be simplified. √806 ≈ 28.390139.

806 is a palindrome in three different bases:

• 11211 BASE 5 because 1(625) + 1(125) + 2(25) + 1(5) + 1(1) = 806
• 1C1 BASE 23 (C is 12 base 10) because 1(23²) + 12(23) + 1(1) = 806
• QQ BASE 30 (Q is 26 base 10) because 26(30) + 26(1) = 806, which follows naturally from the fact that 26 × 31 = 806

806 is the hypotenuse of Pythagorean triple 310-744-806 which is 5-12-13 times 62.

And 806 can be written as the sum of three squares seven different ways:

• 26² + 11² + 3² = 806
• 26² + 9² + 7² = 806
• 25² + 10² + 9² = 806
• 23² + 14² + 9² = 806
• 21² + 19² + 2² = 806
• 21² + 14² + 13² = 806
• 19² + 18² + 11² = 806

# 805 and Level 4

23 × 35 = 805 so we shouldn’t be surprised that 805 is palindrome NN in BASE 34. N is the same as 23 in base 10. Thus NN can be derived from 23(34) + 23(1) = 23(34 + 1) = 23 × 35 = 805. NN obviously is divisible by 11 like all 2 digit palindromes are.

Since 23 = 22 + 1, should we expect that 805 is a palindrome in BASE 22? No, and that is for the same reason that not all multiples of 11 are palindromes.

Finding the factors in today’s puzzle shouldn’t be very difficult, but the last few might be trickier than the rest:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

• 805 is a composite number.
• Prime factorization: 805 = 5 x 7 x 23
• 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 805 has exactly 8 factors.
• Factors of 805: 1, 5, 7, 23, 35, 115, 161, 805
• Factor pairs: 805 = 1 x 805, 5 x 161, 7 x 115, or 23 x 35
• 805 has no square factors that allow its square root to be simplified. √805 ≈ 28.37252

805 is the hypotenuse of a Pythagorean triple:

• 483-644-805, which is 3-4-5 times 161

805 can be written as the sum of three squares four ways:

• 25² + 12² + 6² = 805
• 24² + 15² + 2² = 805
• 20² + 18² + 9² = 805
• 18² + 16² + 15² = 805

# 804 Is There a Pot of Gold at the End of This Rainbow?

Factor Rainbows can be a wonderful way to display the factors of a number. Not only are all the factors listed in order from smallest to greatest, but the factor pairs are joined together with the same color band.

The number 804 has 12 factors so it makes a lovely rainbow with 6 different color bands.

Is there a pot of gold at the end of this factor rainbow? I’ll let you decide the answer to that question.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Finding golden nuggets of information about a number might be less difficult than finding pots of gold.

I always begin the painstaking mining process by looking at the factors of the number:

• 804 is a composite number.
• Prime factorization: 804 = 2 x 2 x 3 x 67, which can be written 804 = (2^2) x 3 x 67
• 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 804 has exactly 12 factors.
• Factors of 804: 1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804
• Factor pairs: 804 = 1 x 804, 2 x 402, 3 x 268, 4 x 201, 6 x 134, or 12 x 67
• Taking the factor pair with the largest square number factor, we get √804 = (√4)(√201) = 2√201 ≈ 28.3548937575

Finding nuggets of information about the number 804 has been a little difficult and disappointing:

• None of 804’s prime factors can be written as 4N+1, so 804 is NOT the hypotenuse of any Pythagorean triples.
• 804 is NOT a palindrome in base 36 or any base less than that.
• 804 is NOT the sum of any consecutive prime numbers.

Even though I did not find any golden nuggets in those places, I kept looking and finally found a couple of gems about the number 804:

804 can be written as the sum of three squares four different ways, and all of those ways have some definition of double in them:

• 28² + 4² + 2² = 804
• 26² + 8² + 8² = 804
• 22² + 16² + 8² = 804
• 20² + 20² + 2² = 804

Stetson.edu also gives us a nugget about the number 804 that may be a bit too heavy for most people to handle: “804 is a value of n for which 2φ(n) = φ(n+1).” That basically means that there are exactly half as many numbers less than 804 that are NOT divisible by its prime factors (2, 3, or 67) as there are numbers less than 805 that are NOT divisible by its prime factors (5, 7, or 23).

I started looking for golden specs about 804 in places that I don’t usually look.

267 + 268 + 269 = 804 so 804 is the sum of 3 consecutive numbers.

As stated before 804 is never the hypotenuse of a Pythagorean triple. However to find all the times it is a leg in a triple will require a lot of labor especially since 804 has so many factors, including 4, and two of its factor pairs have factors where both factors are even.

• 134 × 6 is an even factor pair, so (134 + 6)/2 = 70, and (134-6)/2 = 64. Thus 804 = 134·6 = (70 + 64)(70 – 64) = 70² – 64² .
• 402 × 2 is another even factor pair, so (402 + 2)/2 = 202, and (402 – 2)/2 = 200. Thus 804 = 402·2 = (200 + 2)(200 – 2) = 202² – 200²
• Likewise odd or even sets of factor pairs of any of 804’s factors can also be used to find Pythagorean triples.

So to find all Pythagorean triples that contain the number 804, we will have to find all the times 804 satisfies one of these FOUR conditions:

1. 804 = 2k(a)(b) so that 804 is in the triple 2k(a)(b), k(a² – b²), k(a + b²) OR the triple k(a² – b²), 2k(a)(b), k(a + b²).
2. 804 = 2(a)(b) so that 804 is in the triple 2(a)(b), a² – b², a + b² OR the triple a² – b², 2(a)(b), a + b².
3. 804 = a² – b² so that 804 is in the triple a² – b², 2(a)(b), a + b² OR the triple 2(a)(b), a² – b², a + b².
4. 804 = k(a² – b²) so that 804 is in the triple k(a² – b²), 2k(a)(b), k(a + b²) OR the triple 2k(a)(b), k(a² – b²), k(a + b²).

Let the mining process begin! I’ll list the triples with the shortest legs first and color code each triple according to the condition I used.

• 335-804-871 which used 804 = 2·67(3)(2) to make a triple that is 5-12-13 times 67
• 603-804-1005 which used 804 = 2·201(2)(1) to make a triple that is 3-4-5 times 201
• 804-1072-1340 which used 804 = 268(2² – 1²) to make a triple that  is 3-4-5 times 268
• 804-2345-2479 which used 804 = 2·67(6)(1) to make a triple that is 12-35-37 times 67
• 804-4453-4525, which used 804 = 2(6)(67)
• 804-8960-8996, which used 804 = 70² – 64² or 804 = 4(35² – 32²) to make a triple that is 201-2240-2249 times
• 804-17947-17965, which used 804 = 2(134)(3)
• 804-26928-26940 which used 804 = 12(34² – 33²) to make a triple that is 67-2244-2245 times 12
• 804-40397-40405, which used 804 = 2(201)(2)
• 804-53865-53871 which used 804 = 2·3(134)(1) to make a triple that is 268-17955-17957 times 3
• 804-80800-80804 which used 804 = 202² – 200²  or 804 = 4(101² – 100²) to make a triple that is 201-20200-20201 times 4
• 804-161603-161605, a primitive Pythagorean triple, that used 804 = 2(402)(1)

If you look for a pot of gold at the end of a rainbow, you’re bound to be disappointed. Science/How Stuff Works just had to crush dreams and dispel 10 Myths About Rainbows. Unfortunately a pot of gold being at the rainbow’s end is included on that list. Still I suppose we could still put every golden spec or nugget about 804 into a little pot and call it a pot of gold.

Or if you are as clever and quick as a leprechaun, perhaps you will consider finding Pythagorean triples to be like finding pots of gold.

# 803 From Top to Bottom

8 – 0 + 3 = 11, so 803 is divisible by 11.

• 803 is a composite number.
• Prime factorization: 803 = 11 x 73
• 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 803 has exactly 4 factors.
• Factors of 803: 1, 11, 73, 803
• Factor pairs: 803 = 1 x 803 or 11 x 73
• 803 has no square factors that allow its square root to be simplified. √803 ≈ 3372546

You can solve today’s Level 3 puzzle by starting at the top of the first column, finding the factors of the clues and writing them in the appropriate cells. Then continue to go down that same column, cell by cell, finding factors and writing them down until you reach the bottom. Make sure that both the first column and the top row have each number from 1 to 10 written in them.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Here’s a few more facts about the number 803:

803 is the hypotenuse of a Pythagorean triple:

• 528-605-803 which is 11 times another Pythagorean triple: 48-55-73

803 is the sum of three squares six different ways:

• 27² + 7² + 5² = 803
• 25² + 13² + 3² = 803
• 23² + 15² + 7² = 803
• 21² + 19² + 1² = 803
• 19² + 19² + 9² = 803
• 17² + 17² + 15² = 803

803 is the sum of consecutive prime numbers three different ways. Prime factor 11 is not in any of those ways, but prime factor 73 is in two of them.

• 263 + 269 + 271 = 803, that’s 3 consecutive primes.
• 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 = 803, that’s 9 consecutive primes.
• 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 803, that’s 13 consecutive primes.

803 is a palindrome in two bases. Why are the numbers similar in these two palindromes?

• 30203 BASE 4 because 3(256) + 0(64) + 2(16) + 0(4) + 3(1) = 803
• 323 BASE 16 because 3(16²) + 2(16) + 3(1) = 803

# 802 Pi Day at Smith’s

In the United States tomorrow’s date is written 3-14. Because 3.14 is a famous approximation for π (pi), people all over the country will eat pie to celebrate Pi Day. This afternoon I took a picture of this sign and the pie display at my local Smith’s Food and Drug.

I took that picture right when I walked into the store, but there were no pies on display for National Pi Day.

About 15 minutes later I returned to the display to take another picture. Now there were pies on the table! I told a salesperson who I think worked on the display that I was going to take a picture and put it on my blog. She asked what kind of a blog I wrote. I told her a math blog. She looked puzzled and asked why I would want to put a picture of pies on a math blog. Then she turned around, looked at the display, and said something like, “Oh, now I get it, the number pi.”

How do you choose between apple, cherry, or peach pie? It’s much easier if you choose two and then you can get a free 8 oz. Cool Whip, too. Yummy.

If by chance you prefer pizza pi, here’s a thought from twitter that is often repeated in March:

And here’s some original artwork that displays pi in a way I had never thought of before:

BREAKING: secret of Pi revealed #PiDay pic.twitter.com/Ao8BQp31jd

You can also look here for a million digits of pi.

But pi is not the only interesting number in the world. Every number has its own curiosities. Let me tell you some reasons to get excited about the number 802:

802 is the sum of two squares:

• 21² + 19² = 802

So 802 is the hypotenuse of a Pythagorean triple:

• 80-798-802, which is 2 times another triple: 40-399-401.

It also means something else: Since odd numbers 21 and 19 have no common prime factors, 802 can be evenly divide by 2. Duh. . ., but it also means that unless 802 is also divisible by 5, 13, or 17, its only factors will be 2 and a prime number! Why are those three numbers the only ones I care about? Because they are the only prime number Pythagorean triple hypotenuses less than √802 ≈ 28.3.

Guess what? 5, 13, and 17 do not divide evenly into 802, so 802 is the product of 2 and a prime number which happens to be 401.

• 802 is a composite number.
• Prime factorization: 802 = 2 x 401
• The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 802 has exactly 4 factors.
• Factors of 802: 1, 2, 401, 802
• Factor pairs: 802 = 1 x 802 or 2 x 401
• 802 has no square factors that allow its square root to be simplified. √802 ≈ 28.3196045

Today’s puzzle is number 802 to distinguish it from every other puzzle I’ve made. Writing the numbers 1 – 10 in both the top row and the first column so that the factors and the clues work together as a multiplication table is as easy as pie!

Print the puzzles or type the solution on this excel file: 10-factors 801-806

And here is a little more about the number 802:

802 is the sum of 8 consecutive prime numbers:

• 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 802

802 can also be written as the sum of three squares three different ways:

• 28² + 3² + 3² = 802
• 27² + 8² + 3² = 802
• 24² + 15² + 1² = 802

802 is also a palindrome in two other bases:

• 414 BASE 14 because 4(196) + 1(14) + 4(1) = 802
• 202 BASE 20 because 2(400) + 0(20) + 2(1) = 802

# 801 and Level 1

When it comes to applying our tried and true trick for divisibility by nine to the number 801, zero is just a place holder. Thus, since 81 is divisible by 9, so is 801. Adding up its digits was hardly necessary.

• 801 is a composite number.
• Prime factorization: 801 = 3 x 3 x 89, which can be written 801 = (3^2) x 89
• 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 801 has exactly 6 factors.
• Factors of 801: 1, 3, 9, 89, 267, 801
• Factor pairs: 801 = 1 x 801, 3 x 267, or 9 x 89
• Taking the factor pair with the largest square number factor, we get √801 = (√9)(√89) = 3√89 ≈ 28.301943396.

Would you be surprised to know the following division facts?

• 81 ÷ 3 = 27
• 801 ÷ 3 = 267
• 8001 ÷ 3 = 2667
• 80001 ÷ 3 = 26667 and so forth. The number of 6’s in the quotient is the same as the number of 0’s in the dividend!

Here are some more predictable division facts:

• 81 ÷ 9 = 9
• 801 ÷ 9 = 89
• 8001 ÷ 9 = 889
• 80001 ÷ 9 = 8889 and so forth. You guessed it! The number of 8’s in the quotient is the same as the number of 0’s in the dividend!

Even though you can’t see 81 in this puzzle with all perfect square clues, it isn’t difficult to see where 9 × 9 and 81 belong:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

801 is a palindrome in three bases:

• 1441 BASE 8 because 1(8^3) + 4(8^2) + 4(8) + 1(1) = 801
• 2D2 BASE 17 D is 13 base 10 because 2(289) + 13(17) = 2(1) = 801
• 171 BASE 25 because 1(25²) + 7(25) + 1(1) =801

801 is the sum of two squares:

• 24² + 15² =801

So it follows that 801 is the hypotenuse of a Pythagorean triple:

• 351-720-801 which is 9 times 39-80-89

801 is the sum of three squares TEN ways:

1. 28² + 4² + 1² = 801
2. 27² + 6² + 6² = 801
3. 26² + 11² + 2² =801
4. 26² + 10² + 5² = 801
5. 24² + 12² + 9² = 801
6. 23² + 16² + 4² = 801
7. 22² + 14² + 11² = 801
8. 21² + 18² + 6² = 801
9. 20² + 20² + 1² = 801
10. 17² + 16² + 16² = 801

Stetson.edu gives us this last fun fact:

801 = (7! + 8! + 9! + 10!) / (7 × 8 × 9 × 10).