For the past few years I have been exploring the powers of numbers [squares, cubes etc.]. In a search for patterns in the cubes and to help me sleep I have been learning the cubes in order off by heart.
However, I don't just type in numbers into a calculator to see what the next cube is. Instead, I predict based on what I know about the underlying structure of the cubes.
To introduce you to the complicated structure underlying the sequence of cubes from 1 x 1 x 1 up I challenge you to look at the cubes for numbers in which the last digit is 8, ie.,8 18 28 ....
Below are the first 20 fo these cubes [8 to 198]:
512 195,112 1,259,712 3,944,312
5,832 314,432 1,643,032 4,741,632
21,952 474,552 2,097,152 5,639,752
54,872 681,472 2,628,072 6,644,672
110,592 941,192 3,241,792 7,762,392
The challenge - can you predict the next 20 digits in the 10s column and the next 20 digits in the 100s column WITHOUT calculating the cube.
summing digits till a single digit
1 + 0 = 1
8 + 0 = 8
2 + 7 = 9
6 + 4 = 10 = 1 + 0 = 1
1 + 2 + 5 = 8 = 8
2 + 1 + 6 = 9 = 9
3 + 4 + 2 = 10 = 1 + 0 = 1
5 + 1 + 2 = 8 = 8
7 + 2 + 9 = 18 = 1 + 8 = 9
1 + 0 + 0 + 0 = 1
1 + 3 + 3 + 1 = 8
1 + 7 + 2 + 9 = 18 = 1 + 8 = 9
2 + 1 + 9 + 7 = 19 = 1 + 9 = 10 = 1 + 0 = 1
2 + 7 + 4 + 4 = 17 = 1 + 7 = 8
3 + 3 + 7 + 5 = 18 = 1 + 8 = = 9
So the sequence begins: 1 8 9 1 8 9 1 8 9 1 8 9 1 8 9
Sadly no prize for how the sequence continues.
summing the cubes
1 = 1
1 + 8 = 9
1 + 8 + 27 = 36
1 + 8 + 27 + 64 = 100
1 + 8 + 27 + 64 + 125 = 225
1 + 8 + 27 + 64 + 125 + 216 = 441
1 + 8 + 27 + 64 + 125 + 216 + 343 = 784
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 = 1296
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 = 2025
1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 = 3025
Sequence of sums : 1 9 36 100 225 784 1296 2025 ......
This sequence of sums is also the sequence of something else.
This challenge is slightly more difficult than the one before.
What is your favourite cube ?
How about 23393656, 54439939, 1367631, 37595375 or 1 ?
Or 1331, 373248, 16003008 ?; these cubes provided me with information about the underlying structure of the sequence of ordered cubes
This may be the hardest challenge you have ever had in your life.
highest cubes I have worked out
124,251,499 125,751,501
214,921,799 217,081,801
What numbers are these the cubes for ?
Some help :
970,299 1,030,301
7,880,599 8,120,601
26,730,899 27,270,901
63,521,199 64,481,201
Some real big help
64,000 = 40 cubed, 125,000 = 50 cubed
sure you can work out the others
highest cube I have reached [8.8.10]
by 1st, 2nd, 3rd, ...... 469th cube
99,897,344 ; 100,544,625; 101,194,696; 101,847,563
102,503,232; 103,161,709
This is using a complex set of rules to work out the next cube in the sequence. Essentially I work out the last three digits from my knowledge of the first 200 cubes [basically known by heart] and by estimating the amount of change [which grows gradually in the millions and thousands; by now in the 600,000s], and then by checking if the sum of the digits equals the expected number [1, 8 or 9]. There is more to it than this but this is the essentials.
I'm sure no one will read the previous posting
but may admire the wierdo who worked out the big number [not by cheating; I'm a witness]