1600 How Would You Describe This Horse Race?

Today’s Puzzle:

Do most of the numbers from 1501 to 1600 have 2 factors, 4 factors, 6 factors, or what? A horse race is a fun way to find the answer to that puzzle!

As I’ve done several times before, I’ve made a horse race for this multiple of 100 and the 99 numbers before it. A horse moves when a number comes up with a particular amount of factors. Some of the races I’ve done in the past have been exciting with several lead changes. In other races, one horse ran quite quickly, leaving all other horses in the dust. One previous horse race resulted in a tie.

How will you describe this horse race? Exciting or boring? Surprizing or predictable? Pick your pony and watch the race to the end before you decide on an adjective.

Click here if you would like the Horse Race to be slightly bigger.

1501 to 1600 Horse Race

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Wow! I’ve not seen that happen before! Visually it looks like 4 won the race, but this horse race is really about finding the mode There are two modes, 4 and 8, for the amount of factors for the numbers from 1501 to 1600. It’s about which amount of factors comes up most often for the entire set of numbers, not which one of those occurs first. Thus, for that reason, I would describe the horse race above as deceptive. That horse race looked at the amount of factors five numbers at a time. Here’s what happens if we look at ten numbers at a time:

1501 to 1600 Horse Race (by tens)

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In the second horse race, it is much more clear that the race ends in a tie, and the mode is BOTH 4 and 8.

Prime Factorization of Numbers from 1501 to 1600:

Of those 100 numbers, 38 have square roots that can be simplified; 62 do not.

Factor Trees for 1600:

1600 has MANY possible factor trees. Some are symmetrical; some are not. Here are two nicely shaped ones:

Factors of 1600:

  • 1600 is a composite number and a perfect square.
  • Prime factorization: 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5, which can be written 1600 = 2⁶ × 5².
  • 1600 has at least one exponent greater than 1 in its prime factorization so √1600 can be simplified. Taking the factor pair from the factor pair table below with the same number for both factors, we get
    √1600 = (√40)(√40) = 40. However, you could also use
    √1600 = (√4)(√400) = 2 × 20 = 40,
    √1600 = (√16)(√100) = 4 × 10 = 40, or
    √1600 = (√25)(√64) = 5 × 8 = 40.
  • The exponents in the prime factorization are 6 and 2. Adding one to each exponent and multiplying we get (6 + 1)(2 + 1) = 7 × 3 = 21. Therefore 1600 has exactly 21 factors.
  • The factors of 1600 are outlined with their factor pair partners in the graphic below.

More about the Number 1600:

1600 is the sum of two squares:
32² + 24² = 1600.

1600 is the hypotenuse of two Pythagorean triples:
448-1536-1600, calculated from 32² – 24², 2(32)(24), 32² + 24².
It is also (7-24-25) times 64.
960-1280-1600, which is (3-4-5) times 320.

1600 looks square in some other bases:
1600 = 1(40²) + 0(40) + 0(1), so it’s 100₄₀.
1600 =1(39²) + 2(39) + 1(1), so it’s 121₃₉.
1600 =1(38²) + 4(38) + 4(1), so it’s 144₃₈.
1600 =1(37²) + 6(37) + 9(1), so it’s 169₃₇.

Furthermore, 1600 is a repdigit in base 7:
1600 = 4(7³ + 7² + 7¹ + 7º), so it’s 4444₇.

 

1599 and Level 5

Today’s Puzzle:

Write the numbers 1 to 10 in both the first column and the top row so that this level 5 puzzle will function like a multiplication table. Use logic with every step.

Factors of 1599:

  • 1599 is a composite number.
  • Prime factorization: 1599 = 3 × 13 × 41.
  • 1599 has no exponents greater than 1 in its prime factorization, so √1599 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1599 has exactly 8 factors.
  • The factors of 1599 are outlined with their factor pair partners in the graphic below.

More about the Number 1599:

1599 is the hypotenuse of FOUR Pythagorean triples:
276-1575-1599, which is 3 times (92-525-533),
351-1560-1599, which is 39 times (9-40-41),
615-1476-1599, which is (5-12-13) times 123, and
924-1305-1599, which is 3 times (308-435-533).

1599 is the difference of two squares four different ways:
800² – 799² = 1599,
268² – 265² = 1599,
68² – 55² = 1599, and
40² – 1² = 1599.
Yes, we are just one number away from a perfect square!

1598 See the Logic in This Level 4 Puzzle

Today’s Puzzle:

Put the numbers from 1 to 10 in both the first column and the top row to turn this level 4 puzzle into a multiplication table. The logic in the ten clues is fairly straight-forward. Go ahead give it a try!

Factors of 1598:

  • 1598 is a composite number.
  • Prime factorization: 1598 = 2 × 17 × 47.
  • 1598 has no exponents greater than 1 in its prime factorization, so √1598 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1598 has exactly 8 factors.
  • The factors of 1598 are outlined with their factor pair partners in the graphic below.

More about the Number 1598:

1598 is the hypotenuse of a Pythagorean triple:
752-1410-1598, which is (8-15-17) times 94.

How Much of My $1596 Will the Taxman Get?

Today’s Puzzle is a Game Called Taxman:

Tax season in the United States began just a few days ago. Figuring taxes is no fun, but playing Taxman is.

You see 56 cards below. Imagine that they are 56 envelopes, each containing the dollar amount written on them.

You can take whatever envelope you’d like, but the taxman must be able to take at least one envelope when you do. What envelopes will the taxman take when it is his turn? He will take EVERY available envelope that has a number on it that is a factor of the envelope you took. So if you started off taking envelope 56, the taxman would take envelopes 1, 2, 4, 7, 8, 14, and 28. Thus the taxman would start off getting a whole lot more money than you while also limiting envelopes you can choose later. For example, you wouldn’t be able to pick any prime numbers after taking 56  because the 1 would be gone, and the taxman must be able to take one envelope with every choice you make. When you get to the point that there are no more envelopes that you can take, the taxman gets to take ALL the remaining envelopes. You want the taxman to get less money than you do, and hopefully as little money as possible. I’ve included the factors of each of the numbers at the top of each “envelope” to make the game easier for both you and the taxman. You can print “envelopes” from this excel file Taxman & 1537-1544.  You can have a friend be the taxman as you play or you can play both roles.

How I Played the Game:

I took envelopes as I asked myself these questions:

What is the largest prime number on the table? 53. I take 53, and the only envelope the taxman gets is 1.

What is the largest number that is a prime number squared? 49. I take 49, and the only envelope left for the taxman to take is 7.

What is the largest multiple of 7 that has only one other available factor? 35. I take 35, and the only envelope the taxman can still get is 5.

What is the largest multiple of 5 that has only one other available factor? and so forth.

Here is the order I took the envelopes. It is not the only possible order to use, but it was one in which there were only nine cards at the end of the game for the taxman to claim.

The winner of the game is the one with the most money at the end of the game.

How Do We Know Who Won the Game?

We could add up all the money I got and compare it with all the money the taxman got, but that wouldn’t be much fun. We could add up all the taxman money and subtract it from 1596 to find my total, but I prefer a different way: I remove all the taxman’s money from the table and try to mix envelopes to match mine. I start with my envelope with the most cash, 56 = 8 + 1 + 47, so I put the taxman’s envelopes with those numbers next to my 56.

Matching the “envelopes” up like this will take some time, but it feels like playing a game. It may have taken me as long to play this part of the game as it took me to play Taxman because sometimes I had to remove perfectly good sums in order to use all the taxman’s envelopes in a sum. Here is how I matched the envelopes:

I know that I won the game because I had at least one envelope without a sum next to it. I can determine by  how much I won the game by adding 35 + 33 + 46 + 39 + 27 + 38 + 20 + 30 + 42. That sum tells me I won the game by $310.

How much did the taxman get?
(1596-310)/2 = 643.

How much did I get to keep?
643 + 310 = 953.

A Factor Tree for 1596:

If you know that 40² = 1600, and 1600 – 4 = 1596, we might recognize that 1596 is the difference of two perfect squares: 40² – 2². Then we can also know that
1596 = (40 – 2)(40 + 2) = 38 × 42.

Let’s make a factor tree from the factor pair 38 × 42:

Since 1596 has several factors, it has several other possible factor trees.

Factors of 1596:

  • 1596 is a composite number.
  • Prime factorization: 1596 = 2 × 2 × 3 × 7 × 19, which can be written 1596 = 2² × 3 × 7 ×
  • 1596 has at least one exponent greater than 1 in its prime factorization so √1596 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1596 = (√4)(√399) = 2√399
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1596 has exactly 24 factors.
  • The factors of 1596 are outlined with their factor pair partners in the graphic below.

More about the Number 1596:

The reason that I was dividing up $1596 between me and the taxman is that 1596 is the 56th triangular number. It is the 56th triangular number because
(56)(57)/2 = 1596.

1596 is the difference of two squares in four different ways:

1596 is the sum of the first 15 Fibonacci numbers, which also means it is one number less than the 17th Fibonacci number.

 

 

1595 and Level 2

Today’s Puzzle:

This puzzle is just a multiplication table whose missing factors are not in the usual order. Can you figure out where the factors from 1 to 10 should go?

Factors of 1595:

1595 ends with a 5, so it is divisible by 5.
1 – 5 + 9 – 5 = 0, so 1595 is divisible by 11.

  • 1595 is a composite number.
  • Prime factorization: 1595 = 5 × 11 × 29.
  • 1595 has no exponents greater than 1 in its prime factorization, so √1595 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1595 has exactly 8 factors.
  • The factors of 1595 are outlined with their factor pair partners in the graphic below.

More about the Number 1595:

1595 is the hypotenuse of FOUR Pythagorean triples:
187-1584-1595, which is 11 times (17-144-145),
264-1573-1595, which is 11 times (24-143-145),
957-1276-1595, which is (3-4-5) times 319, and
1100-1155-1595, which is (20-21-29) times 55.

1594 and Level 1

Today’s Puzzle:

Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1594:

  • 1594 is a composite number.
  • Prime factorization: 1594 = 2 × 797.
  • 1594 has no exponents greater than 1 in its prime factorization, so √1594 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1594 has exactly 4 factors.
  • The factors of 1594 are outlined with their factor pair partners in the graphic below.

More about the Number 1594:

1594 is the sum of two squares as well as the double of two squares:
37² + 15² = 1594.
2(26² + 11²) = 1594.

1594 is the hypotenuse of a Pythagorean triple:
1110-1144-1594, calculated from 2(37)(15), 37² – 15², 37² + 15².
It is also 2 times (555-572-797).

Sorting the Factors of 1593

Curious Sort of Factors:

1593 has eight factors. Three-fourths of those factors can be made using some or all of its digits. The other two factors seem to be talking about each other. 27 says 177 has two sevens, and 177 says that 27 has one seven. I’ve sorted them into two categories:

Today’s Puzzle:

Can 1593 be expressed using its four digits exactly once and (), +, -, ×, ÷?
Almost, but not quite:
1593 = 3³ × 59¹. We are not allowed to use the 3 twice.
1593 = 531√9. We can use digits as exponents, but we are not allowed to use the square root symbol.
1593 is not a Friedman number because 1593 = 1593 is the only way we can express it using (), +, -, ×, ÷ and only its own digits as numbers or exponents.

Now try this: Can you express each of 1593’s eight factors using all four of its digits exactly once and only (), +, -, ×, ÷? I’ve done a few of them to get you started. (One of them can’t be done, and there is more than one possibility for some of them.)

1 = 1⁵⁹³
3 =
9 = 3(9-5-1)
27 =
59 =
177 =
531 =
1593 = 1593. That’s all we can do for that one.

Factors of 1593:

  • 1593 is a composite number.
  • Prime factorization: 1593 = 3 × 3 × 3 × 59, which can be written 1593 = 3³ × 59.
  • 1593 has at least one exponent greater than 1 in its prime factorization so √1593 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1593 = (√9)(√177) = 3√177.
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1593 has exactly 8 factors.
  • The factors of 1593 are outlined with their factor pairs in the graphic below.

More about the Number 1593:

Consecutive numbers 1592 and 1593 each have eight factors because they are each a cube times a prime.

1593 is the difference of two squares four different ways:
797² – 796² = 1593,
267² – 264² = 1593,
93² – 84² = 1593, and
43² – 16² = 1593.

1592 One More Valentine

Today’s Puzzle:

I made this mystery level puzzle into one more valentine. Love can seem tricky sometimes, but I hope you enjoy working on it.

Factors of 1592:

  • 1592 is a composite number.
  • Prime factorization: 1592 = 2 × 2 × 2 × 199, which can be written 1592 = 2³ × 199
  • 1592 has at least one exponent greater than 1 in its prime factorization so √1592 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1592 = (√4)(√398) = 2√398
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1592 has exactly 8 factors.
  • The factors of 1592 are outlined with their factor pairs in the graphic below.

More about the Number 1592:

1599 is the difference of two squares two different ways:
399² – 397² = 1592, and
201² – 197² = 1592.

1591 Conversation Heart

Today’s Puzzle:

If this valentine-shaped level 6 puzzle gets kids talking about multiplication, then it will truly be a conversation heart.

Factors of 1591:

  • 1591 is a composite number.
  • Prime factorization: 1591 = 37 × 43.
  • 1591 has no exponents greater than 1 in its prime factorization, so √1591 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1591 has exactly 4 factors.
  • The factors of 1591 are outlined with their factor pair partners in the graphic below.

More about the Number 1591:

1591 is the difference of two squares two different ways:
796² – 795² = 1591, and
40² – 3² = 1591. That means we are only 3², or 9 numbers away from the next perfect square, 40², or 1600.

1590 A Single Rose

Today’s Puzzle:

A single rose can be an elegant expression of affection. This single rose is a level 5 puzzle. Can you find its factors?

Here’s the same puzzle without any added color:

Two Factor Trees for 1590:

There are several possible factor trees for 1590. Here are two of them.

Factors of 1590:

  • 1590 is a composite number.
  • Prime factorization: 1590 = 2 × 3 × 5 × 53
  • 1590 has no exponents greater than 1 in its prime factorization, so √1590 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1590 has exactly 16 factors.
  • The factors of 1590 are outlined with their factor pair partners in the graphic below.

More about the Number 1590:

1590 is the hypotenuse of FOUR Pythagorean triples:
138-1584-1590, which is 6 times (23-264-265),
576-1482-1590, which is 6 times (96-247-265)
840-1350-1590, which is 30 times (28-45-53),
954-1272-1590, which is (3-4-5) times 318.