Back to the original headline topic, the prince and princess puzzle which Gemini 1.5 Pro Experimental 0827 can and does solve (albeit with stuttering in the output), Gemini 1.5 Pro 002 answers as follows (no stuttering, correct answer for the expression for the age of the prince, wrong answer for the age of the princess):
Let 'p' be the present age of the princess and 'r' be the present age of the prince.
Let's break down the word problem piece by piece:
• "when the princess is twice as old as the prince was..." Let's call this past time 't' years ago. At that time, the princess's age was p-t, and the prince's age was r-t. We know p-t = 2(r-t).
• "...when the princess's age was half the sum of their present age." This refers to another point in the past. At this time, the princess's age was (p+r)/2. Since this clause is part of the same sentence, this also occurred 't' years ago. So, p-t = (p+r)/2.
• "A princess is as old as the prince will be..." This refers to a future time when the prince will be a certain age. Let's call the number of years that pass until this time 'x'. The prince's age will be r+x. The princess's current age is equal to the prince's future age: p = r+x
• "...when the princess is twice as old as the prince was when the princess's age was half the sum of their present age." This ties everything together. The prince's age when the princess was half the sum of their present ages was r-t. So the princess's age at a future time, which is equal to the prince's age at that time (r+x), is twice the prince's age 't' years ago: r+x = 2(r-t). And since p = r+x, we also have p = 2(r-t).
Now we have a system of equations:
1. p - t = 2(r - t) => p - t = 2r - 2t => p + t = 2r
2. p - t = (p + r) / 2 => 2p - 2t = p + r => p - 2t = r
3. p = r + x
4. p = 2(r - t) (This is redundant since it's derived from equation 1 and 3, so we can ignore it)
From equation (2), we can express t in terms of p and r:
2t = p - r => t = (p-r)/2
Substitute this into equation (1):
p + (p-r)/2 = 2r
2p + p - r = 4r
3p = 5r
p = (5/3)r
Since ages must be positive integers, we consider cases where 'r' is a multiple of 3.
• If r = 3: p = 5. Then t = (5-3)/2 = 1. This is a valid solution. Princess is 5 and Prince is 3.
• If r = 6: p = 10. Then t = (10-6)/2 = 2. This is also valid. Princess is 10 and Prince is 6.
• General Solution: The relationship p = (5/3)r holds for any r that's a multiple of 3. Therefore, the general solution is:
Prince's age = 3n
Princess's age = 5n
where 'n' is any positive integer.
The prompt itself is as previously reported in How better is the new model (gemini 1.5 pro experimental 0827) - #13 by OrangiaNebula