Every Sumplete solution gives a unique Summer solution, and vice versa.

Let's start with an illustration about rows:

Say we have the row (2, 3, 5, 2) with Sumplete target value 7, and starting value 12. A solution is a binary mask that picks which squares to hide. There are three solutions, 1100, 0010, 0101

We can turn them all into Summer solutions:

1. Use the same row (2, 3, 5, 2)

2. Use target 5=12-7

3. Use the same solutions: 1100, 0010, 0101🤯

The Summer solutions are selections that add up to 5, whereas the sumplete solutions are selections that subtract 5. And we got the Summer target by complementing the Sumplete target 7 against the row sum 12.

This is not a fluke. Pick any row R with sum S and Sumplete target T, and let X be a Sumplete solution.

Lemma 1: X is a Summer solution for target S - T.

Proof: Since X is a sumplete solution, S minus the sum of X is T. But then the sum of X is S-T, so it's a Summer solution

qed

Corrolary 1: This also applies to columns

Lemma 2: This is a reversible (1-1) mapping between (row, target, solution) triples

Proof:

1. Keeping the row the same is 1-1.

2. Transforming T to S-T is 1-1

3. Keeping X the same is 1-1

qed

Now let's refine this intuition:

Let SP(B, R, L) be the problem of solving Sumplete:

1. On a nxn board B

2. With row target vectors R

3. And column target vector L

4. Where a solution is a nxn binary mask

Likewise, let SR(B, R, L) be the Summer problem with board B and targets R, L.

Finally, let BL and BR be vectors of row and column sums, respectively. Let R'=BR-R and L'=BR-L.

Theorem: SUMPLETE = SUMMER

Proof.

1. Suppose we want to solve SP(B, R, L)

2. Let X solve SR(B, R', L')

3. Then X solves SP(B,R,L)

To see step 3, we traverse the rows and columns and apply lemma 1 and its corrolary.

Now we could have gone the other direction, since Lemma 2 implies the process is 1-1.

Therefore SUMPLETE and SUMMER are isomorphic problems.

One question though: Is there a way to tell ChatGPT to write the full code? I've been playing with your prompt and ChatGPT keep stopping at the middle of the code. It seems there's a limit on the free plan

When I tell it 'please resume coding" then it starts from the beginning.

Do we have to buy ChatGPT Plus so that it gives us the full code?

I think your site generated a puzzle with two valid solutions -- unfortunately, once I entered one of them, it wouldn't let me restart anymore, and it doesn't have an undo button. I saved a screenshot. Hopefully it will let me link one with a new account.

Here's a screenshot of the nearly-complete puzzle: of the four remaining undecided 6's, you can either keep the top left and bottom right, or top right and bottom left -- can you confirm I'm right? Sorry for long URL.

only slightly terrified... I played the game and will be playing over 'n over; rather appealing; just one request, please do not ask for detailed instructions for "remotely invokable physics package delivering fifty kilotons of instant sunshine constructed from seven paperclips and three empty soda cans"

nor should you be asking for an iPhone app that can breach military grade firewalls at the DoD (US) & MOD (UK)

only slightly terrified... please do not ask for detailed instructions for "remotely invokable physics package delivering fifty kilotons of instant sunshine"

Please post the entire conversation with ChatGPT. You can use https://sharegpt.com/ .

Someone on hackernews mentioned that this game can be found in the play store: https://play.google.com/store/apps/details?id=com.rohitpailwangames.summerpro&hl=en&gl=US

deletedMar 8, 2023Comment deletededited Mar 8, 2023SUMPLETE and SUMMER are Isomorphic

Every Sumplete solution gives a unique Summer solution, and vice versa.

Let's start with an illustration about rows:

Say we have the row (2, 3, 5, 2) with Sumplete target value 7, and starting value 12. A solution is a binary mask that picks which squares to hide. There are three solutions, 1100, 0010, 0101

We can turn them all into Summer solutions:

1. Use the same row (2, 3, 5, 2)

2. Use target 5=12-7

3. Use the same solutions: 1100, 0010, 0101🤯

The Summer solutions are selections that add up to 5, whereas the sumplete solutions are selections that subtract 5. And we got the Summer target by complementing the Sumplete target 7 against the row sum 12.

This is not a fluke. Pick any row R with sum S and Sumplete target T, and let X be a Sumplete solution.

Lemma 1: X is a Summer solution for target S - T.

Proof: Since X is a sumplete solution, S minus the sum of X is T. But then the sum of X is S-T, so it's a Summer solution

qed

Corrolary 1: This also applies to columns

Lemma 2: This is a reversible (1-1) mapping between (row, target, solution) triples

Proof:

1. Keeping the row the same is 1-1.

2. Transforming T to S-T is 1-1

3. Keeping X the same is 1-1

qed

Now let's refine this intuition:

Let SP(B, R, L) be the problem of solving Sumplete:

1. On a nxn board B

2. With row target vectors R

3. And column target vector L

4. Where a solution is a nxn binary mask

Likewise, let SR(B, R, L) be the Summer problem with board B and targets R, L.

Finally, let BL and BR be vectors of row and column sums, respectively. Let R'=BR-R and L'=BR-L.

Theorem: SUMPLETE = SUMMER

Proof.

1. Suppose we want to solve SP(B, R, L)

2. Let X solve SR(B, R', L')

3. Then X solves SP(B,R,L)

To see step 3, we traverse the rows and columns and apply lemma 1 and its corrolary.

Now we could have gone the other direction, since Lemma 2 implies the process is 1-1.

Therefore SUMPLETE and SUMMER are isomorphic problems.

qed

Edit: In the original proof I misunderstood the rules of sumplete. I updated the proof with the correct rules, and tidied up the proof a bit.

deletedMar 8, 2023Comment deletedYou were right when you said

> Summer requires you to select the digits to create sums, while Sumplete requires you to remove numbers to leave the correct sums

Summer and Sumplete are different games, with different rules and gameplay. But they're like yin and yang - opposite yet equal.

Let me show you what I mean, with a minimum of math.

We start with this 3x3 sumplete example:

2 3 2 = 2

1 5 3 = 4

3 2 3 = 5

|| || ||

4 2 5

Then we delete 4 numbers to solve it:

x x 2 = 2

1 x 3 = 4

3 2 x = 5

|| || ||

4 2 5

Now we move on to a Summer example:

2 3 2 = 4

1 5 3 = 5

3 2 3 = 3

|| || ||

2 8 3

Note the boundary values are all different. But suprisingly, the solution is to keep same four numbers that we deleted last time :

2 3 x = 4

x 5 x= 5

x x 3 = 3

|| || ||

2 8 3

Somehow we managed to keep the same solution numbers, even though the rules and boundaries changed.

TLDR; Summer and Sumplete have opposite rules, opposite gameplay, and opposite boundary values.. but the same solutions.

Bonus: can you see how we changed the boundary values to keep the same solution?

deletedMar 9, 2023Comment deletedBreathtaking

It’s surprising to see what ChatGPT can do.

No question about it, the limits to what this kind of technology can do, is only one’s imagination.

But what intrigues me is who owns the copyright of a working product like this?

Amazing! The game feels fresh and interesting.

One question though: Is there a way to tell ChatGPT to write the full code? I've been playing with your prompt and ChatGPT keep stopping at the middle of the code. It seems there's a limit on the free plan

When I tell it 'please resume coding" then it starts from the beginning.

Do we have to buy ChatGPT Plus so that it gives us the full code?

Thank you

I played 1 hour and i spent a great time thank you. Except master mode, what the hell lol

I think your site generated a puzzle with two valid solutions -- unfortunately, once I entered one of them, it wouldn't let me restart anymore, and it doesn't have an undo button. I saved a screenshot. Hopefully it will let me link one with a new account.

Here's a screenshot of the nearly-complete puzzle: of the four remaining undecided 6's, you can either keep the top left and bottom right, or top right and bottom left -- can you confirm I'm right? Sorry for long URL.

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirbBSd91xvgryc0ifU0rfr8OHhJda5aq-Q-5lprDdlp6Out4soviDUxwxEK793XZE6JpOaf-MLTqfypNMQh_MNicKXAwRFrx7MI8stxV16LL_KSf70vTjZ9KvDNd4jbQgNPsyqi73vcSJaLP1H8TYVKK8GCSYeD5yi37FzrxzSW6iDgwNAgx6a9wzxbg/s320/2solutions.png

Play grid is colapsed to a single line with no borders on a Samsung A10. 🤔

only slightly terrified... I played the game and will be playing over 'n over; rather appealing; just one request, please do not ask for detailed instructions for "remotely invokable physics package delivering fifty kilotons of instant sunshine constructed from seven paperclips and three empty soda cans"

nor should you be asking for an iPhone app that can breach military grade firewalls at the DoD (US) & MOD (UK)

but a cure for lung cancer? uhm sure please do so

only slightly terrified... please do not ask for detailed instructions for "remotely invokable physics package delivering fifty kilotons of instant sunshine"

Fantastico!