1458 Tangrams Can Be A Pot of Gold

A Tangram Puzzle

Tangrams are seven puzzle pieces that can form a square but can also be made into many different people, places, and things. A lot of stress is going on in the world right now, but since tomorrow is Saint Patrick’s Day, we can still find a little pot of gold at the end of the rainbow!

I made this pot of gold on Desmos using points and equations. If you cut it apart, will you be able to put it back together again?

What other things can you make from those seven tangram shapes?

And what about that rainbow I mentioned? The number 1458 makes a lovely factor rainbow.

A Factor Rainbow for 1458:

Factors of the number 1458:

  • 1458 is a composite number.
  • Prime factorization: 1458 = 2 × 3 × 3 × 3 × 3 × 3 × 3, which can be written 1458 = 2 × 3⁶.
  • 1458 has at least one exponent greater than 1 in its prime factorization so √1458 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1458 = (√729)(√2) = 27√2
  • The exponents in the prime factorization are 1 and 6. Adding one to each exponent and multiplying we get (1 + 1)(6 + 1) = 2 × 7 = 14. Therefore 1458 has exactly 14 factors.
  • The factors of 1458 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1458:

2 is a prime factor of 1458 exactly one time, so there are NO ways that 1458 can be written as the difference of two squares.

2 and 3 are the only primes appearing in its prime factorization, so 1458 is NEVER the hypotenuse of a Pythagorean triple.

Nevertheless, since there are three different ways that 1458 = 2(a)(b), where a > b, there are three ways that 1458 is a leg in a Pythagorean triple:
1458-531440-531442, calculated from 2(729)(1), 729² – 1², 729² + 1²
1458-59040-59058, calculated from 2(243)(3), 243² – 3², 243² + 3²
1458-6480-6642, calculated from 2(81)(9), 81² – 9², 81² + 9²

Why can’t we get a Pythagorean triple from 1458 = 2(27)(27)? I’m sure you can figure out that one yourself.

 

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1036 Look, Look to the Rainbow

Finian’s Rainbow is a wonderful movie to enjoy on Saint Patrick’s Day. One of its songs reminds us to “Look, look to the rainbow”.

If you look to this rainbow, you will find all the factors of 1036:

There is a simple symmetry in every rainbow. There is also symmetry in palindromes which are numbers, words, or sentences that read the same forward or backward.

1036 demonstrates that symmetry when it is written in some other bases:
It’s repdigit 4444 in BASE 6 because 4(6³ + 6² + 6¹ + 6⁰) = 4(216 + 36 + 6 + 1) = 4(259) = 1036,
It’s 232 in BASE 22 because 2(22²) + 3(22) + 2(1) = 1036,
1M1 in BASE 23 (M is 22 base 10) because 23² + 22(23) + 1 = 1036, and
SS in BASE 36 (S is 28 base 10) because 28(36 + 1) = 28(37) = 1036

1036 is also the hypotenuse of a Pythagorean triple:
336-980-1036 which is 28 times 12-35-37

  • 1036 is a composite number.
  • Prime factorization: 1036 = 2 × 2 × 7 × 37, which can be written 1036 = 2² × 7 × 37
  • 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 × 2 × 2 = 12. Therefore 1036 has exactly 12 factors.
  • Factors of 1036: 1, 2, 4, 7, 14, 28, 37, 74, 148, 259, 518, 1036
  • Factor pairs: 1036 = 1 × 1036, 2 × 518, 4 × 259, 7 × 148, 14 × 74, or 28 × 37,
  • Taking the factor pair with the largest square number factor, we get √1036 = (√4)(√259) = 2√259 ≈ 32.18695

There wasn’t a pot of gold at the end of our factor rainbow, but there is one here at the end of this post. It’s a level 5 puzzle, but it isn’t too difficult, so see if you can find all the factors that make the puzzle function like a multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043