I challenge you to the following test: how good is ChatGPT at reasoning? In order to evaluate the ability of ChatGPT to perform inferences, I suggest you ask the bot to solve the problem of 12 balls, which I have described below. Since we want to evaluate how “smart” ChatGPT really is, I recommend to choose the more precise conversation style. I also advise you to be patient with the esteemed bot and carefully guide him, first through the ramifications of the first case, then through those of the second.
After you obtained a satisfactory answer, please share it in the comment section of this post. I look forward to seeing what responses you got! After that, I will post the answer I myself had received from the honorable bot in April 2023.
The problem of 12 balls:
We are given 12 balls that are identical in shape, appearance, and mass, except for one ball that has a different mass from the others. We do not know whether that ball is lighter or heavier than the others. Using a scale and up to 3 weighings, we are asked to identify the ball that is different and how is different, i.e. it’s heavier or lighter than the benchmark balls.
Solution:
We number the balls from 1 to 12 and separate them into 3 groups of 4 balls each: {a1, a2, a3, a4}; {b5, b6, b7, b8}; and {c9, c10, c11, c12}.
The first weighing:
We compare {a1, a2, a3, a4} with {b5, b6, b7, b8}.
If the scale is balanced, then we have case 1.
If the scale is unbalanced, then we have case 2.
Case 1
If when weighing {a1, a2, a3, a4} versus {b5, b6, b7, b8} the scale is balanced, then the different ball is in the group {c9, c10, c11, c12} and any one of the first 8 balls can be used as benchmark.
We break the group {c9, c10, c11, c12} in two subgroups: {c9, c10, c11} and {c12}.
The second weighing:
1.2. We compare {c9, c10, c11} with 3 benchmark balls that we have identified after the first weighing.
If the scale is balanced, then the different ball is c12 and we proceed to the third weighing 1.3a.
If the scale is unbalanced, then the different ball is among the balls c9, c10 and c11; the direction the scale tipped tells us how that ball is different – heavier or lighter – than a benchmark ball. We proceed then to the third weighing 1.3b.
The third weighing:
1.3a. We compare the ball c12 with a benchmark ball.
The direction the scale tips tells us how the ball c12 is different – heavier or lighter – than a benchmark ball.
1.3b. We break the group {c9, c10, c11} in two subgroups: one with any two of these balls and one with the remaining ball.
We compare the balls from the first subgroup one against the other.
If the scale tipped, then the tipping direction together with the conclusion drawn at step 1.2. help us identify the different ball and how that ball is different – heavier or lighter – than a benchmark ball. Thus, if we knew from step 1.2 that the different ball is heavier, then the ball that moved down is the different ball; if we knew that the different ball is lighter, then the ball moving up is the different ball.
If the scale is balanced, then the ball from the second subgroup is the different ball and the conclusion drawn at step 1.2. tells us how that ball is different – heavier or lighter. Thus, if at step 1.2. the group that ball was in moved the scale down, the different ball is heavier; if the group that ball was in moved the scale up, the different ball is lighter.
Case 2
If when weighing {a1, a2, a3, a4} versus {b5, b6, b7, b8} the scale is unbalanced, then the different ball is one of the first 8 balls and any of the balls {c9, c10, c11, c12} can be used as benchmark. It is important to remember which way the scale has tipped.
We swap the balls a4 and b5 and replace the other 3 balls in the second subgroup with 3 benchmark balls.
The second weighing:
2.2. We compare {a1, a2, a3, b5} with {a4, c, c, c}.
If the scale is balanced, then the balls a1, a2, a3, a4, b5 are benchmark balls and the different ball is among the balls b6, b7, b8. Also, from the first weighing we know how the balls b6, b7, b8 compare to the benchmark balls (heavier or lighter). We proceed to the third weighing 2.3a.
If the scale is unbalanced, then the different ball is among the balls a1, a2, a3, a4, b5 and the balls b6, b7, b8 can be used as benchmarks. We proceed to the third weighing 2.3b.
The third weighing:
The next steps are determined by the direction in which the scale tipped during the first two weighings.
2.3a. We break the group {b6, b7, b8} in two subgroups: one with any two of these balls and one with the remaining ball. Next we apply the reasoning described in step 1.3b.
2.3b. This step is particularly dependent on the direction in which the balance was tipped at the first weighing.
- 2.3b.1. If in step 2.2. the scale tilted in the same direction as at the first weighing, then the different ball is one of balls a1, a2, a3 and, depending on the tilt, we know how it is – heavier or lighter – compared to the benchmark balls. We break the group {a1, a2, a3} into two subgroups: one with any two of these balls and one with the remaining ball. Next, we apply the reasoning described in step 1.3b.
- 2.3b.2. If in step 2.2. the scale tipped in the opposite direction to that of the first weighing, then the different ball is one of balls a4, b5. We compare any of the balls a4, b5 with a benchmark ball.
If the scale is balanced, then the different ball is the unmeasured ball and the direction of the scale’s tilt at the first weighing tells us how this ball is – heavier or lighter – compared to a benchmark ball.
If the scale is unbalanced, then the different ball is the measured ball and the direction the scale tilted at the first weighing tells us how this ball is – heavier or lighter – compared to a benchmark ball.
Note: After I wrote this article, I found out that this problem has several solutions. You can find another solution here:
https://codingnconcepts.com/puzzle/12-balls-weight-puzzle/
The ‘intelligence’ is a measure of solving new problems, not old ones, or apply solutions of old problems to a new space. To this measure, finding a solution for a famous puzzle is not a very good test of intelligence. The current models of GenerativeAI are not at a stage where one can call them ‘intelligent’ in this manner, though they show ability to do new things.
This is the solution given to me by ChatGPT. I hadn’t had the patience to check if solution is right or not:
Certainly! The puzzle you’ve presented is a classic one, and it’s known as the 12 Balls Weight Puzzle. Let’s break down the solution step by step:
Numbering the Balls:
First, we’ll assign a number to each of the 12 balls: 1, 2, 3, …, 12.
Dividing into Groups:
Divide the 12 balls into three groups of four balls each: Group 1, Group 2, and Group 3.
Initial Weighing:
Place Group 1 balls on one side of the scale and Group 2 balls on the other side.
There are three possible outcomes:
1. If the scale balances, the defective ball is in Group 3.
2. If Group 1 is heavier than Group 2, the defective ball is either in Group 1 (heavier) or Group 2 (lighter).
3. If Group 2 is heavier than Group 1, the defective ball is either in Group 2 (heavier) or Group 1 (lighter).
Further Steps Based on Initial Weighing:
Let’s explore each case:
Case 1: Scale Balances (Group 3 contains the odd ball):
Weigh any two balls from Group 3 against each other.
If they balance, the odd ball is Ball 12. Weigh it against any other ball to determine if it’s heavier or lighter.
If Group 3 is heavier, it contains a heavy ball (let’s call it Ball X). Weigh Balls 9 and 10 against each other:
If they balance, Ball 11 is the odd heavy ball.
If not, the heavier of Balls 9 and 10 is the odd heavy ball.
If Group 3 is lighter, it contains a light ball (let’s call it Ball Y). Weigh Balls 9 and 10 against each other:
If they balance, Ball 11 is the odd light ball.
If not, the lighter of Balls 9 and 10 is the odd light ball.
Case 2: Group 1 is Heavier:
Weigh Balls 5, 6, and 1 against Balls 7, 2, and 12:
If they balance, either Ball 8 is the odd light ball or Balls 3 or 4 are the odd heavy balls.
Weigh Balls 3 and 4 against each other to find the odd heavy ball.
If Balls 7, 2, and 12 are heavier, either Ball 2 is the odd heavy ball or Balls 5 or 6 are the odd light balls.
Weigh Balls 5 and 6 against each other to find the odd light ball.
If Balls 7, 2, and 12 are lighter, either Ball 7 is light or Ball 1 is heavy.
Weigh Ball 7 against any other ball to determine the odd ball.
Case 3: Group 2 is Heavier:
Weigh Balls 1, 2, and 5 against Balls 3, 6, and 12:
If they balance, either Ball 4 is the odd light ball or Balls 7 or 8 are the odd heavy balls.
Weigh Balls 7 and 8 against each other to find the odd heavy ball.
If Balls 1, 2, and 5 are heavier, either Ball 2 is the odd heavy ball or Balls 5 or 6 are the odd light balls.
Weigh Balls 5 and 6 against each other to find the odd light ball.
If Balls 1, 2, and 5 are lighter, either Ball 1 is light or Ball 7 is heavy.
Weigh Ball 7 against any other ball to determine the odd ball.