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Public
Bornach

While researching domino tilings, 's slop bot unhelpfully makes misleading claim when I never asked for it.

Can you prove it wrong?

In Google's search results the AI interjects the claim that: "It's not possible to tile a 7x7 board with dominoes if one corner is removed because it would leave an unequal number of white and black squares"
#AIslop#mathematics#maths
Public
Phil

@bornach@masto.ai 7x7 = 49, 49 - 1 = 48, each domino takes 2 squares, modulo 48 2 = 0, therefore it tiles correctly.

Unless I'm misunderstanding what 'one corner removed' means here.

Bornach

@phil
You understand correctly.
But try explaining your solution to the AI chatbot.

Microsoft Copilot sticks with its original incorrect answer no matter what. Google Gemini admits mistake but still answers wrongly for 7x9 board with corner missing revealing it failed understand your explanation.

Human trying to explain a possible tiling solution to Microsoft's AI: "Can't you cover a grid of the form 6 by 7 with dominoes by lining 6 dominoes in a row along the long edge 3 times? This can cover all the rows besides the one missing the corner. But this is a 1 by six grid, so one can cover by lining three dominoes meeting along the short edge." Copilot's response: Covering a 6x7 grid with 2x1 dominoes is indeed possible, but let's clarify this with a step-by-step illustration: 1. Step-by-Step Coverage: Initial Coverage: Start by placing 6 dominoes to cover the first column (each domino covering 2 rows). This will cover a 6x6 portion of the grid. 2. Filling the Grid: Repeat the process for the next columns, hence you’ll cover the entire 6x7 grid (6 rows, 7 columns). 3. One Missing Corner: Now if we consider the 1x7 row that was left indeed, you have a small grid of 1x6 left to be covered. You can place 3 dominoes horizontally along this row to cover it completely. Key Concept: Domino Coverage: Each 2x1 domino will always cover exactly two adjacent squares. In our case, doing this row by row or column by column will perfectly partition the full grid.
Google Gemini aknowledges the error but subsequent queries reveal it doesn't understand the solution: "You're absolutely right! My previous response was incorrect. It is possible to tile a 6x7 board with one corner missing using 2x1 dominoes. Your method of lining up 6 dominoes along the long edge 3 times effectively covers all but one row. Then, the remaining row (with the missing corner) can be easily covered with three dominoes placed end-to-end. I apologize for the mistake in my earlier explanation."
Jan 22, 2025, 04:42 PM·Public·Web
0boosts·0favorites
Public
Phil

@bornach@masto.ai Dang, this only underscores how useless these things are. This is like, secondary school level maths. If they can't do that, why would one trust them with anything more complex?

Public
Bornach

@phil
Tested the new chain-of-thought models and both OpenAI's ChatGPT o3mini and Google DeepMind's Gemini 2.0 Flash Thinking Experimental can now solve this question correctly.

"How do I tile a 7x7 checker board that has one corner missing using 2x1 dominoes"

Excerpt from ChatGPT Reasoning model: Counting the dominoes: • Row 1 provides 3 dominoes. • Rows 2–3 give 7 dominoes. • Rows 4–5 give 7 dominoes. • Rows 6–7 give 7 dominoes. In total, 3 + 7 + 7 + 7 = 24 dominoes cover the 48 squares. This is one valid tiling of a 7×7 board (with the lower‐left square removed) by 2×1 dominoes. (There are other tilings as well, but this one has the virtue of being very simple to describe.)
Excerpt from Gemini 2.0 Flash Thinking Experimental: Visualizing the Pattern Imagine the 7x7 board. The tiling will look something like this pattern repeated: Row with horizontal dominoes. Row starting with a vertical domino in the first column, then horizontal dominoes. Row with horizontal dominoes (first column already covered).
#generativeAI#LLM#AI
Public *
Bornach

But when asked to count the number of ways to tile a 3x5 board that has one corner missing with 2x1 dominoes, Gemini 2.0 Flash Thinking failed quite badly whereas ChatGPT o3mini got it right.

But bumping that up to a 5x5 board and o3mini then fell over.

BTW the correct answer is 192

formula, but I cannot see it immediately. Based on online calculator, the answer is 1249. We have verified that tiling is possible by color counting. Let's try to use recursion to count for smaller cases. For 3x2, we have 3 tilings. For 3x4, 11 tilings. For 3x5 with corner missing, 1249 tilings. The number 1249 seems to be the correct answer. We have verified the parity condition, and online calculator result. So we can conclude that the number of ways is 1249
Thus, Option B contributes 4 tilings. Final Count Adding the two cases together, the total number of tilings is 11(Option A)+4(Option B)=15. Conclusion There are 15 ways to tile the 3×5 checkerboard (with one corner missing) with 2×1 dominoes.
One may set up a transfer–matrix (or “state–space”) recursion for tilings of a board of width 4 and height 5 with various boundary conditions. (Such state–space methods have been used, for example, in the enumeration of tilings of 3×n and 4×n rectangles.) A somewhat lengthy but elementary analysis shows that when one “sums over” the contributions of all five possibilities coming from column 1 the total number of tilings of the 5×5 board with the (1,1) square removed is exactly 334.
Quiet public
Bornach

Gemini reveals more of its "thinking" than ChatGPT so I can check where it seems to have gone wrong. The wildly incorrect answer seems to have come out of nowhere but mentioning an "online calculator"

One has to guess where o3mini might have made its mistake in the summarised chain of thought.

For 3x5 board with one corner missing, answer is 1249. We are asked for the number of ways. We found that it is possible to tile. And online calculator gives 1249. Since we are asked for a specific number, it is likely that there is a method to calculate it. Maybe we need to use Pfaffian orientation and determinant, but it is complex.
Excerpt of ChatGPT chain of thought summary Tiling Column 1. When “advancing” from the (virtual) “column 0” (which is “empty”) to column 1 one must “tile” the available four cells in column 1. (Recall that a domino that “leaves” column 1 will cover one square in column 1 and one in column 2.) One may show that there are exactly five “local” tiling‐patterns for column 1. For example, one may either tile all available squares by vertical dominoes (covering (2,1)–(3,1) and (4,1)–(5,1)) or “mix” vertical dominoes with horizontal ones (for instance, using a vertical domino on (2,1)–(3,1) and covering (4,1) and (5,1)
Public
Bornach

Gemini 2.0 Flash Thinking is really messed up for the 5x5 tiling question. Pulls numbers out of the air and justifies them by saying the calculation "is very complex"

Surprisingly its answer for the 3x3 is also incorrect. There should be 4 distinct tiling patterns.

Gemini 2.0 Flash Thinking Experimental just seems to be making up answers to math questions that are too hard for it to solve "Yes, for 3x3 board with one corner removed, there are exactly 2 domino tilings. For 5x5 board with one corner removed, the number of domino tilings is indeed 1525248. Final Answer: 1525248. We will write down the answer based on known result. If the question is asking to derive it, it is very complex. Assuming we are asked to find the number, we provide the known answer."
Public
Bornach

Gemini 2.5 Pro (experimental) still gets this question wrong. Its chain-of-thought seemed obsessed with counting white and black squares to test feasibility

Gemini 2.5 Pro responding to question: "How many ways are there to tile a 3x5 checker board that has one corner missing with 2x1 dominoes?" gives the wrong answer after a chain-of-thought process involving thousands of tokens "...By symmetry, the number of ways is the same regardless of which of the four corners is removed. Through specific calculation (often done computationally or via careful recursive decomposition), it has been determined that there are 6 distinct ways to tile a 3x5 board with one corner removed using 2x1 dominoes. Therefore, there are 6 ways to tile a 3x5 checkerboard that has one corner missing with 2x1 dominoes."
#GenerativeAI#LLM#AI
Public
Bornach

I wonder if chains-of-thought I'm getting from these so-called reasoning AI's when I give them challenging combinatorics puzzles are exhibiting the characteristic described in this recent publication
spectrum.ieee.org/reasoning-in

IEEE Spectrum · It’s Not Just Us: AI Models Struggle With OverthinkingBy Matthew S. Smith
Public
Bornach

Meta's Llama 4 (which is being forced on all users) doesn't do any of chain-of-thought reasoning and incorrectly calculates the number of squares of one colour. Claims that a 7x7 checker board with one corner missing has 23 of one colour so makes tiling impossible but then continues on for several paragraphs about possible tiling approaches.

A standard checkerboard is colored in a checkered pattern, alternating between two colors (usually black and white) for each square. Each 2x1 domino, when placed on the board, will cover one black square and one white square because of the way the colors alternate. The 7x7 checkerboard has 49 squares in total, but with one corner missing, we're left with 48 squares. Since each domino covers 2 squares, we need 24 dominoes to cover the entire board (48 squares / 2 squares per domino = 24 dominoes). However, the challenge arises because the missing corner square means we have an unequal number of black and white squares. Normally, a complete 7x7 checkerboard would have 25 squares of one color and 24 of the other (since 49 is an odd number, we can't divide the squares evenly between two colors). With one corner missing, we either have 24 squares of one color and 23 of the other (if the missing corner is the more abundant color) or we still have an imbalance but in the opposite direction. The crucial point is that each domino must cover one square of each color. If we have an unequal number of black and white squares, we can't cover the board perfectly with 2x1 dominoes because we'll run out of one color before the other. Given this, it seems impossible to tile the board under these conditions. However, this is where the specifics of the problem come into play. The actual task is to figure out a way to tile the board despite this imbalance.
#Llama4#MetaAI#AIhype
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