Today’s Puzzle:

1770 = 30 · 59.
1770 = (60 · 59)/2.

That means 1770 is a triangular number. If we have 59 envelopes numbered 1 to 59, and each envelope contained the amount of money on the outside of the envelope, we would have $1770 in cash at stake. In this game, the TAXMAN wants to take as much money as he can get, but you control how much he can take: Can you allow him to get as little as possible? 

You can play Taxman easily with these printable Taxman “envelopes” and Taxman Scoring Calculator because each “envelope” lists all the factors of the envelope number. Your first selection should be the biggest prime number on the board because then the only envelope the taxman can get on that turn is the 1 envelope. The Taxman must be able to take at least one envelope on every turn. Try to make it so he can only get one or at most two envelopes on each turn. When it is no longer possible for you to take an envelope that allows the Taxman to take at least one envelope, too, the taxman gets ALL the rest of the envelopes. You win if you can keep more than half of your cash. Good luck!

Factors of 1770:

• 1770 is a composite number.
• Prime factorization: 1770 = 2 × 3 × 5 × 59.
• 1770 has no exponents greater than 1 in its prime factorization, so √1770 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 1770 has exactly 16 factors.
• The factors of 1770 are outlined with their factor pair partners in the graphic below.

More About the Number 1770:

As mentioned earlier 1770 is the 59th triangular number because 59(60)/2 = 1770.

1770 is also the 30th hexagonal number because 30(2·30-1) = 1770. (Every hexagonal number is also a triangular number.)

I’ve made images of hexagonal numbers before, but this time I wanted to make one using this hexagon template:

1770 is the hypotenuse of a Pythagorean triple:
1062-1416-1770, which is (3-4-5) times 354.

1770 is repdigit, UU, in base 58 because
30(58) + 30(1) = 30(58 + 1) = 30(59) = 1770.