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1738 Little Surprises Around Every Turn

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Today’s Puzzle:

Is this mystery-level puzzle easy or difficult? I’m not saying. Place all the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues. If you use logic to solve the puzzle, you will probably find some surprises around every turn.

Factors of 1738:

It may surprise you that 1738 is divisible by 11. Why is it? Because
1 – 7 + 3 – 8 = -11, a multiple of 11, which makes 1738 divisible by 11.

  • 1738 is a composite number.
  • Prime factorization: 1738 = 2 × 11 × 79.
  • 1738 has no exponents greater than 1 in its prime factorization, so √1738 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 1738 has exactly 8 factors.
  • The factors of 1738 are outlined with their factor pair partners in the graphic below.

More About the Number 1738:

Maybe you will be surprised by some of these number facts about the number 1738.

As mentioned in the table, 1738 = 2 × 11 × 79. That means 1738 is the short leg in the Pythagorean triple calculated from 2(79)(11), 79² – 11², 79² + 11².

1738 is the sum of eleven consecutive numbers:
153 + 154 + 155 + 156 +157 + 158 + 159 + 160 + 161 + 162 + 163 = 1738.

1738 is the sum of four consecutive numbers:
433 + 434 + 435 + 436 = 1738.

And because four is an even number:
436² – 435² + 434² – 433² = 1738.
Surprised?

1738 is also the sum of the 44 consecutive numbers from 18 to 61.

Consequently,
61² – 60² + 59² – 68² + 57² – 56² + 55² – 54² + 53² – 52² + 51² – 50² + 49² – 48² + 47² – 46² + 45² – 44² + 43² – 42² + 41² – 40² + 39² – 38² + 37² – 36² + 35² – 34² + 33² – 32² + 31² – 30² + 29² – 28² + 27² – 26² + 25² – 24² + 23² – 22² + 21² – 20² + 19² – 18² = 1738. I bet you weren’t expecting that!

But the biggest surprise about 1738 is something I learned from OEIS.org: 1738 is in an equation that uses every digit from 1 to 9 exactly one time:
9 Different Digit Equation

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Factors and Facts for the Year 2021

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Countdown to 2021:

Here is a countdown you can use on New Year’s Eve to bring in 2021:

Countdown to 2021

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It’s Almost Like 2021 Is Looking in a Mirror!

Squaring means multiplying a number by itself. Something unusual happens when you square 2021 and you also square its reverse, 1202. Complete the two multiplication problems below to see for yourself!

There are only 50 numbers less than 10000 with this same property. They are all listed in the table below:

For each pair of numbers in the table, the smaller number was listed first, but if you look carefully you will notice 2001, 2011, 2012, 2021, and 2022 in this century and 2101, 2102, 2111, and 2121 in the next.

Math Facts about the Number 2021:

2021 Tiny Squares

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Here’s a Magic Square showing 2021:

2021 is the difference of two squares in two different ways:
45² – 2² = 2021 and
1011² – 1010² = 2021.

2021 is not the hypotenuse of any Pythagorean triples, but it is a leg in four:
180-2021-2029, calculated from 2(45)(2), 45² – 2², 45² + 2²,
2021-43428-43475, which is (43-924-925) times 47,
2021-47472-47515, which is 43 times (47-1104-1105)
2021-2042220-2042221, from 1011² – 1010², 2(1011)(1010), 1011² + 1010².

Factors of 2021:

The previous year had many negative factors, although it had just as many positive ones. I can predict that 2021 will have only four negative factors to go along with its four positive ones. I think we will see a lot of the negative factors at the beginning of the year, but hopefully, the positive ones will be more evident as the year goes on.

  • 2021 is a composite number.
  • Prime factorization: 2021 = 43 × 47.
  • 2021 has no exponents greater than 1 in its prime factorization, so √2021 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 2021 has exactly 4 factors.
  • The factors of 2021 are outlined with their factor pair partners in the graphic below.

Those factors can help you with quadratic equations like these:

Mathematical Tweets about 2021:

Here are tweets about the number 2021 in the order that I’ve seen them. If I see more, I’ll add more.

 

1556 Stacks Up Nicely!

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What is special about the number 1556?

What makes 1556 stack up?

From OEIS.org we learn that
2² + 3² + 5² + 7² + 11² + 13² + 17² + 19² + 23² =  1556.
Yes, that’s the sum of the squares of the first nine prime numbers.
Those perfect squares can be stacked on top of each other as I illustrate in the graphic below:

Factors of 1556:

1556 (and every other whole number whose last two digits are 56) is divisible by 4:

  • 1556 is a composite number.
  • Prime factorization: 1556 = 2 × 2 × 389, which can be written 1556 = 2² × 389.
  • 1556 has at least one exponent greater than 1 in its prime factorization so √1556 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1556 = (√4)(√389) = 2√389.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1556 has exactly 6 factors.
  • The factors of 1556 are outlined with their factor pair partners in the graphic below.

More about the Number 1556:

1556 is the sum of two squares:
34² + 20² = 1556.

1556 is the hypotenuse of a Pythagorean triple:
756 -1360-1556, which is 4 times (189-340-389).
It can also be calculated from 34² – 20², 2(34)(20), 34² + 20².

1556 is also the difference of two squares:
390²  – 388²  = 1556.