With no puzzles this week, I decided to try last week's puzzle as applied to next year (2025).
1. Knowing that the rectangular prism with the largest volume for a given internal diagonal is a cube, I find that the sides of the cube should be 2025/√3 = 1169.134, the maximum volume (noninteger x,y,z) is (2025/√3)³ = 1598060439.6271. To find the closest volume to this maximum for integer x,y,z, I plotted a spreadsheet table where x steps down from 1169, y steps up from 1169, and the table is filled with the possible z values, z = √(2025² - x² - y²) and I look for integer values of z in the table. I find 2 candidates:
x * y * z = 1194 * 1158 * 1155 = 1596963060, and
x * y * z = 1183 * 1144 * 1180 = 1596955360
The first of these two is slightly bigger, so my answer to the first puzzle is:
x * y * z = 1194 * 1158 * 1155 = 1596963060 <<======================
2. 2025 has quite a long Collatz sequence, 157 numbers. A Collatz sequence that contains 2025 must therefore have more than 157 numbers. I started with numbers less than 2025 and worked down, looking for one with a Collatz sequence of more than 157 numbers. I looked only at those Collatz sequences longer than 157 and it appears to me that the smallest number with 2025 in its Collatz sequence is, ... 2025. A surprise considering the vast number of numbers in the Collatz sequences for numbers between 1 and 2025.
I'm a little confused by the term "rectangular prism". Can I assume that this is the same as a rectangular cuboid, i.e. a right rectangular prism?
With no puzzles this week, I decided to try last week's puzzle as applied to next year (2025).
1. Knowing that the rectangular prism with the largest volume for a given internal diagonal is a cube, I find that the sides of the cube should be 2025/√3 = 1169.134, the maximum volume (noninteger x,y,z) is (2025/√3)³ = 1598060439.6271. To find the closest volume to this maximum for integer x,y,z, I plotted a spreadsheet table where x steps down from 1169, y steps up from 1169, and the table is filled with the possible z values, z = √(2025² - x² - y²) and I look for integer values of z in the table. I find 2 candidates:
x * y * z = 1194 * 1158 * 1155 = 1596963060, and
x * y * z = 1183 * 1144 * 1180 = 1596955360
The first of these two is slightly bigger, so my answer to the first puzzle is:
x * y * z = 1194 * 1158 * 1155 = 1596963060 <<======================
2. 2025 has quite a long Collatz sequence, 157 numbers. A Collatz sequence that contains 2025 must therefore have more than 157 numbers. I started with numbers less than 2025 and worked down, looking for one with a Collatz sequence of more than 157 numbers. I looked only at those Collatz sequences longer than 157 and it appears to me that the smallest number with 2025 in its Collatz sequence is, ... 2025. A surprise considering the vast number of numbers in the Collatz sequences for numbers between 1 and 2025.
Seems like the semicircle puzzle should have been on the 15th. Or on Star Wars Day.