### Today’s Puzzle:

A couple of days ago on Twitter, I saw an interesting puzzle posed by Sunil Singh @Mathgarden.

This is probably the greatest algebra problem that can be given to ALL students. The bridges represent SUMS of adjacent fields. There are 8 UNKNOWNS! Without using Algebra–which is accessible to MS students–this problem is hard. Algebra makes it easier…but how?;) pic.twitter.com/C7L7TJgBrt

— Sunil Singh (@Mathgarden) May 11, 2021

After I found one of several of its solutions, I wondered if I could add a bridge that would use all nine numbers from 1 to 9 in the solution, so I tweaked it. I decided to move the puzzle to the ocean when I added that extra bridge.

I was able to solve this problem using logic and addition facts, rather than algebra. Try solving it yourself. If you want to see any of the steps I used to solve the puzzle, scroll down to the end of the post.

### Factors of 1649:

- 1649 is a composite number.
- Prime factorization: 1649 = 17 × 97.
- 1649 has no exponents greater than 1 in its prime factorization, so √1649 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 1649 has exactly 4 factors.
- The factors of 1649 are outlined with their factor pair partners in the graphic below.

### More About the Number 1649:

1649 is the sum of two squares in TWO different ways:

32² + 25² = 1649, and

40² + 7² = 1649.

1649 is the hypotenuse of FOUR Pythagorean triples:

399-1600-1649, calculated from 32² – 25², 2(32)(25), 32² + 25²,

560-1551-1649, calculated from 2(40)(7), 40² – 7², 40² + 7²,

776 1455 1649, which is (8-15-**17**) times **97**, and

1105 1224 1649, which is **17** times (65-72-**97**).

### Some Logical Steps to Solve Today’s Puzzle:

I found four different ways to write the solution but they are all rotations or reflections of each other. Here are the steps to one of those four ways:

1st Step: The biggest number that can be used is 9. Since every island must be included in at least two sums, neither 8 nor 9 can be an addend; they both must be sums. 7 must be an addend in their sums because adding any number to 7 will yield 8, 9, or some larger forbidden number. Thus 8 and 9 are bridges that connect to island 7.

I chose to write 7, 8, and 9 in the top right section, but 8 and 9 could change places with each other. I could have also chosen to write those numbers in the bottom left area.

2nd Step: **1 **+ 7 = 8, and 7 + **2** = 9.

3rd Step: 1 + 2 = **3**.

4th Step: The last island must be the smallest remaining number (**4**) because the smallest remaining number can’t be the sum of a bigger number and the number on either adjacent island.

Final Step: 1 + 4 = **5**, and 4 + 2 = **6**.

Did you enjoy this puzzle? How did my steps compare to the steps that you took?

Please, check the comments for another solution.

Fun puzzle! I started with a different assumption, and found a different solution. I made 7, 8, and 9 all bridges coming off the 6 island, and everything else fell into place. The other islands of course had to be 1, 2, 3 — with 1 and 2 NOT sharing a bridge.

Oh yes! I followed what you wrote and was able to solve it using your assumption, too! (Not on my first try, mind you, because 1 and 2 shared a bridge on my first attempt, but as soon as I put 9 on the center bridge, and 7 and 8 on the other bridges coming off the 6 island, everything worked beautifully.)