## Poll

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35 members have voted

unJon
Joined: Jul 1, 2018
• Posts: 3132
Thanks for this post from:
March 1st, 2021 at 8:23:53 AM permalink
Quote: Gialmere

Here's an easy Monday puzzle...

Did it quick in my head but looks like:

153
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
teliot
Joined: Oct 19, 2009
• Posts: 2288
Thanks for this post from:
March 1st, 2021 at 8:46:15 AM permalink
Quote: Gialmere

Here's an easy Monday puzzle...

153 -- which, coincidentally, is the smallest integer > 1 that is the sum of the cubes of its digits. 153 = 1^3 + 5^3 + 3^3. There are three more positive integers > 1 with this property. Bonus easy Monday question -- What are they?

Bonus bonus question -- how many fish did Simon Peter catch (New Testament)?
Last edited by: teliot on Mar 1, 2021
Poetry website: www.totallydisconnected.com
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 5150
Thanks for this post from:
March 1st, 2021 at 9:11:30 AM permalink

By 891, one of 1, 8, 9 is in the code, but by 849, neither 8 nor 9 are in the code, so 1 is in the code, but not the third digit
By 317, 1 is not the second digit; therefore, 1 is the first digit
By 793, either 9 is the second digit or 3 is the third digit, but by 849, 9 is not in the code, so 3 is the third digit
By 725, either 7 or 5 is the second digit, but by 793, since 3 is in the code, 7 is not, so 5 is the second digit
The code is 153

unJon
Joined: Jul 1, 2018
• Posts: 3132
March 1st, 2021 at 9:23:12 AM permalink
Quote: ThatDonGuy

By 891, one of 1, 8, 9 is in the code, but by 849, neither 8 nor 9 are in the code, so 1 is in the code, but not the third digit
By 317, 1 is not the second digit; therefore, 1 is the first digit
By 793, either 9 is the second digit or 3 is the third digit, but by 849, 9 is not in the code, so 3 is the third digit
By 725, either 7 or 5 is the second digit, but by 793, since 3 is in the code, 7 is not, so 5 is the second digit
The code is 153

You can solve also without reference to the 849 box.

ETA:

Comparing 725 and 793 box you can eliminate 7. From 317 you then know 3 and 1 are correct but not in those positions. Looking at 793 you can put 3 in last position. Then looking at 891 leaves the 1 in first position. Finally looking at 725 you see that 5 has to be the middle digit.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
gordonm888
Joined: Feb 18, 2015
• Posts: 3446
March 1st, 2021 at 9:49:18 AM permalink
Quote: teliot

153 -- which, coincidentally, is the smallest integer > 1 that is the sum of the cubes of its digits. 153 = 1^3 + 5^3 + 3^3. There are three more positive integers > 1 with this property. Bonus easy Monday question -- What are they?

Bonus bonus question -- how many fish did Simon Peter catch (New Testament)?

370
371
407

At first, I thought the question was "prime numbers that are the sum of their digits," and I couldn't find any.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
teliot
Joined: Oct 19, 2009
• Posts: 2288
March 1st, 2021 at 11:34:07 AM permalink

0
1
153
370
371
407

Ok, here is a really easy follow-up question. The list in the spoiler gives all six solutions to "sum of the cubes of the digits equals the number."

My follow-up question is to find all integers that minimize the difference when it is greater than 0, that is:

minimize |(sum of cube of digits of number) - number| > 0

There are six solutions when this difference is equal to 0. I found 12 solutions to the next smallest difference.
Poetry website: www.totallydisconnected.com
charliepatrick
Joined: Jun 17, 2011
• Posts: 2511
March 1st, 2021 at 12:55:26 PM permalink
Quote: teliot

...I found 12 solutions to the next smallest difference.

I could only find 11 - so still looking!
N: 12 C: 9 diff: 3
N: 30 C: 27 diff: 3
N: 31 C: 28 diff: 3
N: 32 C: 35 diff: 3
N: 255 C: 258 diff: 3
N: 365 C: 368 diff: 3
N: 437 C: 434 diff: 3
N: 474 C: 471 diff: 3
N: 747 C: 750 diff: 3
N: 856 C: 853 diff: 3
N: 1799 C: 1802 diff: 3
teliot
Joined: Oct 19, 2009
• Posts: 2288
March 1st, 2021 at 1:03:40 PM permalink
Quote: charliepatrick

I could only find 11 - so still looking!

N: 12 C: 9 diff: 3
N: 30 C: 27 diff: 3
N: 31 C: 28 diff: 3
N: 32 C: 35 diff: 3
N: 255 C: 258 diff: 3
N: 365 C: 368 diff: 3
N: 437 C: 434 diff: 3
N: 474 C: 471 diff: 3
N: 747 C: 750 diff: 3
N: 856 C: 853 diff: 3
N: 1799 C: 1802 diff: 3

I confess to miscounting! You got it. Here are the solutions with the difference <= 10:

1, 0
153, 0
370, 0
371, 0
407, 0
12, 3
30, 3
31, 3
32, 3
255, 3
365, 3
437, 3
474, 3
747, 3
856, 3
1799, 3
22, 6
226, 6
372, 6
1079, 6
10, 9
11, 9
125, 9
216, 9
417, 9
566, 9
675, 9
766, 9
872, 9
873, 9
962, 9
963, 9
Poetry website: www.totallydisconnected.com
teliot
Joined: Oct 19, 2009
• Posts: 2288
March 1st, 2021 at 1:05:44 PM permalink
Okay, which leads me to a cute little theorem-ette that is on the slightly non-trivial side (but not that non-trivial), related to the 153 question.

Let N > 0 be any integer. Let S be the sum of the cubes of its digits. Show that the difference (N-S) is always divisible by 3.

For example, N = 1263. S = 1^3 + 2^3 + 6^3 + 3^3 = 1 + 8 + 216 + 27 = 252. N - S = 1011= 3*337.
Poetry website: www.totallydisconnected.com
charliepatrick
Joined: Jun 17, 2011
• Posts: 2511
March 1st, 2021 at 1:27:42 PM permalink
We can split any number into its constituent digits and consider each one in turn. So a number such as 543 look at the difference between 500 and 53, 40 and 43, 3 and 33. If each of these is divisible by 3, then the total will.

So the approach is to show it for one digit numbers, then two digit numbers, etc.

Consider 0<N<10. N3-N = N * (N2-1) = N * (N-1) * (N+1). One these must be a multiple of 3.

Now consider a number <100 which is 10M+N. Then we would like M3-10M to be a multiple of 3. This is the same as (M3-M)-9*M. The former (M3-M) is divisible by 3 (as per above) and 9*M is also divisible by 3. Hence the difference is divisible by 3. This proves the case for two digit numbers.

Similar logic will apply to any other digit in the number since it will split into (L3-L)-999....999*L.

Hence it applies to all integers.