"The answer to the weekly puzzle is pi-reminiscent: 3, 1, -4.
The mean clue told me I needed to consider negatives.
The median clue told me that one of the values was 3 or -3.
The range clue meant that I was restrained to quite a small range of 7.
The square of the product meant that the size of the three original numbers' product (ignoring the sign) was 12, which was handy since I knew that I already had a factor whose absolute value was 3.
I then revisited the clues and 3,1,-4 sort of dropped into my lap (yes, that final line of explanation is a bit dissatisfying but I'm just being honest).
"
Wil,
Thursday, August 1, 2024
"The numbers are -3, -1 and 4. It took me more than a moment to work it out: since doubling the numbers doesn't double the mean, at first my thought was that the middle number was 0, and they were -n, 0 and n, maybe -3½, 0 and 3½ because the range is 7. But that doesn't work because the product is 0 and we need the product to be 12 or -12. So try -3, 1, 4: everything OK now except that the mean doesn't stay the same when the numbers are doubled; the mean must be 0. So try -3, -1, 4"
Paul,
Thursday, August 1, 2024
"If I square all the numbers the new median is nine - requires a 3 in the answer.
The square of the product of the numbers is 144 - product of the three numbers is 12 - 1 a 3 and a 4
The range of the numbers is seven - lowest possible range of positive numbers is 1 - 8 -> put thinking cap on
Tried using a 0 to help things out but the product requirement ruled that out so only option was to go negative
Tried -3,1,4 didn't meet the mean rule but then -3,-1,4 did.
"
Leonard,
Friday, August 2, 2024
"If the mean stays the same after doubling, we can deduce that the three numbers span the origin. That is, at least one is positive and at least one is negative.
The median clue tells us that one of the numbers is either 3 or -3.
The square clue tells us that the product of the three numbers is either 12 or -12.
We can factor twelve into integer triplets as follows: {1, 1, 12}, {1, 2, 6}, {1, 3, 4} and {2, 2, 3}. The first two are eliminated for not containing a 3.
There is no way to get a range of 7 with {2, 2, 3}, so we focus on {1, 3, 4}.
Adding the appropriate signs, we get two solutions: {1, 3, -4} and {-1, -3, 4}.
"
Rick,
Sunday, August 4, 2024
"Here is my answer to the August 2024 podcast puzzle:
I am thinking of three numbers. If I double them, the mean stays the same.
Hence, if the numbers are a, b, and c:
(a+b+c)/3=(2a+2b+2c)/3 or
a+b+c=0
This means that some of the numbers (I am assuming integers) are positive and some are negative.
If I square all the numbers, the new median is nine.
Therefore, one of the numbers is three or negative three. Because nine is the median, this number is also not the largest or smallest, in absolute value.
The square of the product of the numbers is 144.
This means that the product of the numbers is 12 or -12. Since one of the numbers is three or negative three, and it is the middle number, the other numbers must be, in absolute value, one and four.
The range of the values is seven.
I am not sure why this clue is needed. There are no other numbers that satisfy the other criteria other than one, three, and four. Now, which ones are negative and which ones are positive? I see two possible solutions:
Negative three, negative one, and four, or
Negative four, one, and three,.
Chris,
Thursday, August 1, 2024
"The answer to the weekly puzzle is pi-reminiscent: 3, 1, -4.
The mean clue told me I needed to consider negatives.
The median clue told me that one of the values was 3 or -3.
The range clue meant that I was restrained to quite a small range of 7.
The square of the product meant that the size of the three original numbers' product (ignoring the sign) was 12, which was handy since I knew that I already had a factor whose absolute value was 3.
I then revisited the clues and 3,1,-4 sort of dropped into my lap (yes, that final line of explanation is a bit dissatisfying but I'm just being honest). "
Wil,
Thursday, August 1, 2024
"The numbers are -3, -1 and 4. It took me more than a moment to work it out: since doubling the numbers doesn't double the mean, at first my thought was that the middle number was 0, and they were -n, 0 and n, maybe -3½, 0 and 3½ because the range is 7. But that doesn't work because the product is 0 and we need the product to be 12 or -12. So try -3, 1, 4: everything OK now except that the mean doesn't stay the same when the numbers are doubled; the mean must be 0. So try -3, -1, 4"
Paul,
Thursday, August 1, 2024
"If I square all the numbers the new median is nine - requires a 3 in the answer.
The square of the product of the numbers is 144 - product of the three numbers is 12 - 1 a 3 and a 4
The range of the numbers is seven - lowest possible range of positive numbers is 1 - 8 -> put thinking cap on
Tried using a 0 to help things out but the product requirement ruled that out so only option was to go negative
Tried -3,1,4 didn't meet the mean rule but then -3,-1,4 did. "
Leonard,
Friday, August 2, 2024
"If the mean stays the same after doubling, we can deduce that the three numbers span the origin. That is, at least one is positive and at least one is negative.
The median clue tells us that one of the numbers is either 3 or -3.
The square clue tells us that the product of the three numbers is either 12 or -12.
We can factor twelve into integer triplets as follows: {1, 1, 12}, {1, 2, 6}, {1, 3, 4} and {2, 2, 3}. The first two are eliminated for not containing a 3.
There is no way to get a range of 7 with {2, 2, 3}, so we focus on {1, 3, 4}.
Adding the appropriate signs, we get two solutions: {1, 3, -4} and {-1, -3, 4}. "
Rick,
Sunday, August 4, 2024
"Here is my answer to the August 2024 podcast puzzle:
I am thinking of three numbers. If I double them, the mean stays the same.
Hence, if the numbers are a, b, and c:
(a+b+c)/3=(2a+2b+2c)/3 or
a+b+c=0
This means that some of the numbers (I am assuming integers) are positive and some are negative.
If I square all the numbers, the new median is nine.
Therefore, one of the numbers is three or negative three. Because nine is the median, this number is also not the largest or smallest, in absolute value.
The square of the product of the numbers is 144.
This means that the product of the numbers is 12 or -12. Since one of the numbers is three or negative three, and it is the middle number, the other numbers must be, in absolute value, one and four.
The range of the values is seven.
I am not sure why this clue is needed. There are no other numbers that satisfy the other criteria other than one, three, and four. Now, which ones are negative and which ones are positive? I see two possible solutions:
Negative three, negative one, and four, or
Negative four, one, and three,.
Both seem to satisfy all the clues. "