How Much of My $1540 Will The Taxman Get?

The Taxman Game

You might think that the picture below is just the numbers from 1 to 55 arranged in a 5×11 rectangle.

It’s really PLAY MONEY, one bill in each denomination from 1 to 55. If you add up the total value of all this play money, you will get 1540 because the sum of the numbers from 1 to 55 is 1540.

Most of that 1540 in play money can be yours to keep if you outsmart THE TAXMAN.

You can take any bill that you want, but The Taxman has to get paid every time you do. The Taxman will get ALL the available factors of the bill you take. If there isn’t at least one bill on hand that is a factor of your chosen bill, then you can’t take that bill. When none of the bills left are factors of any of the other bills that are left, The Taxman gets ALL the remaining bills. The value of those bills could add up quickly, and that’s how The Taxman might outsmart you.

You can print this excel file to play the Taxman game: Taxman & 1537-1544 It has all the play money bills from 1 to 100, and all the factors are typed on the top of the bills!

The factors at the top of each bill make it possible to play the game even if you don’t know how to multiply or divide yet. My granddaughter learned to play the game sometime around her eighth birthday. Before tackling 55 bills, she learned what to do for a smaller number of bills. I told her:
If 1 is the only bill, The Taxman wins.
If 1, 2 are the only bills, you take the 2, The Taxman gets the 1, and you win.
If 1, 2, 3 are the only bills, the best you can do is a draw (1 + 2 = 3),
But if there are at least four bills, it is possible to win every time.

She played the game over and over adding one more bill to the game each time she played. She enjoyed it very much and played it over 20 times until bedtime required her to stop. Each time she played we talked about how she would arrange the cards in a rectangle. For example, if she used cards 1 – 6, she would look at the factors of 6 and decide on a 2 × 3 rectangle or a 1 × 6 rectangle.

Knowing I wanted to write this post about the 55th triangular number (1540), I had her jump to using 55 bills the next day, but I helped her with it a little. We didn’t want to just win the game, we wanted to make it so The Taxman would get as little money as possible. Here is the final order of what we took and what The Taxman got.

Laying it out like that made it easy to have do-overs when needed.

A Way to Keep Score When Playing Taxman:

When it was time to score the game, instead of adding up all the numbers to see if we took more than The Taxman, we found smaller sums that equaled as many of our choices as we could as illustrated below. For example,  47 + 6 = 53, so there was no need to add 47 + 6 to The Taxman’s total or to add 53 to ours.

I found it enjoyable finding all those sums, but it isn’t necessarily easy for a child to do. After arranging all those sums on the table, we were able to see that we scored
35 + 39 + 38 + 12 + 30 + 50 + 24 + 48 + 42 = 318 more than The Taxman did.

We can use algebra to figure out exactly how much of our 1540 in play money we get to keep.
If The Taxman got X of the play money, we kept X + 318 of it, and
X + X + 318 = 1540,
2X + 318 = 1540,
2X + 318 – 318 = 1540 – 318,
2X = 1222,
X = 611, that’s what The Taxman got.
We kept 611 + 318 = 929 of our play money.

What percentage of the play money did The Taxman get?
611/1540 ≈ 0.40 which is 40%. Wow, that’s a lot!

Factors of 1540:

  • 1540 is a composite number.
  • Prime factorization: 1540 = 2 × 2 × 5 × 7 × 11, which can be written 1540 = 2² × 5 × 7 × 11
  • 1540 has at least one exponent greater than 1 in its prime factorization so √1540 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1540 = (√4)(√385) = 2√385
  • 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 1540 has exactly 24 factors.
  • The factors of 1540 are outlined with their factor pair partners in the graphic below.

Factor Cake for 1540:

1540 has lots of prime factors with 11 being the greatest so it makes a lovely factor cake:

What Kind of Shape Is 1540 in?

I know that 1540 = 55×56÷2 so, 1540 is the 55th triangular number. Guess what? Since 55 is also a triangular number (55 = 10×11÷2), we could have arranged the play money bills in a triangle like this instead of a 5 × 11 rectangle.

1540 is not only the 55th triangular number, but it is also the sum of the first 20 triangular numbers:

1+3+6+10+15+21+28+36+45+55+66+78+91+105+120+136+153+171+190+210 = 1540.

If you stack those twenty triangular numbers from smallest to biggest, you will get something shaped like a tetrahedron. That’s why we say that 1540 is the 20th tetrahedral number. We can also use a formula for the 20th tetrahedral number:
20(21)(22)/6 = 1540.

1540 is also the 28th hexagonal number. (All hexagonal numbers are also triangular numbers. That’s why it’s the 2(28)-1 = 55th triangular number.) It is the 28th hexagonal number because 2(28²) – 28 = 1540.

1540 is the 20th decagonal number because 4(20²) – 3(20) = 1540.

I didn’t make a visual, but 1540 is also the19th centered nonagonal number.
That’s because
3 × 19 = 57 and (57 – 2)(57 – 1)/2 = 1540.

1540 is also the hypotenuse of a Pythagorean triple triangle:
924-1232-1540 which is (3-4-5) times 308.

1540 is quite the shape-shifter number, isn’t it!

1330 is the 19th tetrahedral Number

The product of any three consecutive counting numbers is always divisible by 6. Why? Because one of the numbers has to be divisible by 3 and at least one number has to be divisible by 2. Dividing the product by 6 always results in a  tetrahedral number. 1330 is a good example:
(19)(20)(21)/6 = 1330

Since the first number in that product was 19, we know that 1330 is the 19th tetrahedral number, and it is the sum of the first 19 triangular numbers:

You can count all 1330 little green squares if you want in the graphic above if you choose.

Here are some more facts about the number 1330:

  • 1330 is a composite number.
  • Prime factorization: 1330 = 2 × 5 × 7 × 19
  • 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 × 2 × 2 × 2 = 16. Therefore 1330 has exactly 16 factors.
  • Factors of 1330: 1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330
  • Factor pairs: 1330 = 1 × 1330, 2 × 665, 5 × 266, 7 × 190, 10 × 133, 14 × 95, 19 × 70, or 35 × 38
  • 1330 has no square factors that allow its square root to be simplified. √1330 ≈ 36.46917

1330 is the sum of the twenty-two prime numbers from 17 to 107. How cool is that?

1330 is the hypotenuse of a Pythagorean triple:
798-1064-1330 which is (3-4-5) times 266

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!

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

  • 816 is a composite number.
  • Prime factorization: 816 = 2 x 2 x 2 x 2 x 3 x 17, which can be written 816 = 2⁴ 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:

680 What Would Happen If Ten-Frames Looked Like This?

680 is a number made using only even digits. (There’s much more about 680 at the end of the post.)

Numbers ending in 0, 2, 4, 6, or 8 are even. Numbers ending in 1, 3, 5, 7 or 9 are odd. Those two simple concepts are not always easy for young children to understand.

Sometimes we teach rhymes to children to help them know the difference:

  • 0, 2, 4, 6, 8; being EVEN is just great.
  • 1, 3, 5, 7, 9; being ODD is just fine.

Still students in early grades struggle with the concepts of odd and even.

Another seemingly simple concept is what pairs of numbers add up to ten. That concept also isn’t as easy for children to understand as adults might think.

Donna Boucher is an elementary school math interventionist with many years experience. Besides many other topics, she is an expert on teaching adding and subtracting to first and second graders. Here are a couple of her tweets with links to her site:

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and

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Free ten-frame flash cards are available on her site to help students learn addition and subtraction facts. What a powerful way for students to learn! She also has Halloween/Thanksgiving ten-frames for sale at Teachers Pay Teachers.

As I read her post about how to use the ten-frame flash cards I wondered what would happen if we followed her instructions EXACTLY, but the ten-frames looked like this:

7 mitten ten-frame
Children would still learn how to add and subtract, but would they also instinctively learn the difference between odd and even numbers?

Would they figure out for themselves that adding two even numbers or adding two odd numbers ALWAYS makes an even number? Or that adding an odd number and an even number together ALWAYS makes an odd number? Or would changing the ten-frames not make any difference at all? Will the mitten ten-frames only make a difference if the parent/teacher/tutor talks about the odd and even numbers?

I don’t know the answer to those questions, but I think the idea is worth trying. I’ve made Mitten Ten-Frames for all the numbers from 0 to 10. The “empty” frames have outlines of mittens to help children know if a left or a right mitten belongs there. The mitten ten-frames don’t have a second border to guide in cutting them out, so the flashcards might not look as good as Donna Boucher’s, but they should still work as flashcards. Follow Donna Boucher’s instructions exactly. If you use the mitten ten-frames, please add a comment to let me know whether or not they make any difference helping students learn the properties of odd and even numbers.

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Here’s more about the number 680:

1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² = 680.

Because 5, 17, and 85 are some of its factors, 680 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factor of each triple?

  • 104-672-680
  • 288-616-680
  • 320-600-680
  • 408-544-680

680 the 15th tetrahedral number. Stetson.edu tells us that it is also the smallest tetrahedral number that can be made by adding two other tetrahedral numbers together, specifically the sum of the 10th and the 14th tetrahedral numbers equals this 15th tetrahedral number as shown below:

  • (10)(11)(12)/6 = 220
  • (14)(15)(16)/6 = 560
  • 220 + 560 = 680
  • (15)(16)(17)/6 = 680

Finally, here is the factoring information for 680:

  • 680 is a composite number.
  • Prime factorization: 680 = 2 x 2 x 2 x 5 x 17, which can be written 680 = (2^3) x 5 x 17
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 680 has exactly 16 factors.
  • Factors of 680: 1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680
  • Factor pairs: 680 = 1 x 680, 2 x 340, 4 x 170, 5 x 136, 8 x 85, 10 x 68, 17 x 40, or 20 x 34
  • Taking the factor pair with the largest square number factor, we get √680 = (√4)(√170) = 2√170 ≈ 26.0768096.

 

455 and Level 6

I don’t mean to sound greedy, but if there were 13 days of Christmas instead of only 12, my true love would give me 455 gifts instead of only 364. That’s because the sum of the first 13 triangular numbers is 455. Come on, that’s 91 more gifts. Funny thing, 91 is one of the factors of 455. Also, I know I’m not the first person to notice that (13 x 14 x 15)/6 = 455. As I’m sure you can see, 455 is a fabulous tetrahedral number.

455 Puzzle

Print the puzzles or type the factors on this excel file:  12 Factors 2015-04-06

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  • 455 is a composite number.
  • Prime factorization: 455 = 5 x 7 x 13
  • 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 445 has exactly 8 factors.
  • Factors of 455: 1, 5, 7, 13, 35, 65, 91, 455
  • Factor pairs: 455 = 1 x 455, 5 x 91, 7 x 65, or 13 x 35
  • 455 has no square factors that allow its square root to be simplified. √455 ≈ 21.3307

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455 Logic