$1653, Fake Primes, and Taxman’s “Rigged” Counting

Today’s Puzzle:

If you’ve ever played Taxman or watched someone else play it, you know that the Taxman gets all available factors of each card you take. You can only take a card if the Taxman can also take at least one card on that turn. When the game is over, the Taxman gets ALL the leftover cards.

1653 is the 57th triangular number so if all the cards in the puzzle were envelopes containing dollar amounts indicated on the outside of the envelope, there would be $1653 at stake.

Many people start Taxman by taking the largest prime number followed by the largest prime number squared. What if a person claimed long before the game started that the only way he or she could lose is if the game is rigged? Most people have never played this game and might believe that claim especially if they perceive that the person making that claim is pretty good at math. Besides given the opportunity, won’t the Taxman take far more than his fair share just so he can spend it on frivolous projects? The fact that all remaining cards at the end of the game go to the Taxman will make the rigged claim seem even more plausible. Furthermore, what if “our math whiz” confidently called out his or her first two number choices, fake prime number 57 followed by perfect square 49? (57 is a composite number, but it often fools people into thinking it’s prime. You could call it a fake prime because it looks like a prime number but isn’t actually prime. Other fake primes are 51, 87, and 91.)

For today’s puzzle, I would like you to play this Taxman game with the mistaken assumption that 57 and 51 are prime numbers. Of course, the Taxman will know better. It will still be possible to win, but it will be much more difficult.

You can print the cards to play Taxman from this file: 10 Factors 1650-1660 with Taxman Scoring Calculator. You might choose to have someone else be the Taxman while you stand far enough away not to be able to see the factors listed on the top of the cards. Whether you are close to the cards or far away, don’t allow yourself any do-overs.

I’ve included a taxman scoring calculator in that excel file. Only enter numbers under “My Cards” and “Taxman Cards”. The rest of the data will auto-populate. You win if your tax rate is less than 50%. I would be very interested to know if you win or if you lose.

Factors of 1653:

  • 1653 is a composite number.
  • Prime factorization: 1653 = 3 × 19 × 29.
  • 1653 has no exponents greater than 1 in its prime factorization, so √1653 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 1653 has exactly 8 factors.
  • The factors of 1653 are outlined with their factor pair partners in the graphic below.

More About the Number 1653:

1653 is the hypotenuse of a Pythagorean triple:
1140-1197-1653, which is (20-21-29) times 57.

1653 = 29 × 57.
1653 is the 29th hexagonal number, and
1653 is the 57th triangular number.

All hexagonal numbers are also triangular numbers. Can you look at the graphic above and see why that’s true? The broken line that I drew might be helpful. It separates the odd numbers from the even ones.

1653 is the 57th triangular number because (57)(58)/2 = 1653.
It is the 29th hexagonal number because 2(29²) – 29 = 1653.