[math-fun] Cumulative sums and hits
Hello Math-Fun, We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process. If, at some stage, the concatenation starts with the string K, we have a hit. K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29 Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34 Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,... HIT! Are all integers potential hits? Best É.
I am pretty sure that nothing with a zero as the second digit can be a hit. On Sat, Nov 2, 2019, 4:42 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-Fun,
We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process.
If, at some stage, the concatenation starts with the string K, we have a hit.
K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29
Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34
Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,...
HIT!
Are all integers potential hits? Best É.
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10, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9, 1 10, hit. On Sat, Nov 2, 2019 at 1:59 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am pretty sure that nothing with a zero as the second digit can be a hit.
On Sat, Nov 2, 2019, 4:42 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-Fun,
We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process.
If, at some stage, the concatenation starts with the string K, we have a hit.
K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29
Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34
Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,...
HIT!
Are all integers potential hits? Best É.
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How does 110 begin with "11"? On Sat, Nov 2, 2019, 5:08 PM Tomas Rokicki <rokicki@gmail.com> wrote:
10, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9, 1 10, hit.
On Sat, Nov 2, 2019 at 1:59 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am pretty sure that nothing with a zero as the second digit can be a hit.
On Sat, Nov 2, 2019, 4:42 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-Fun,
We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process.
If, at some stage, the concatenation starts with the string K, we have a hit.
K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29
Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34
Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,...
HIT!
Are all integers potential hits? Best É.
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Oops. On Sat, Nov 2, 2019 at 2:14 PM Allan Wechsler <acwacw@gmail.com> wrote:
How does 110 begin with "11"?
On Sat, Nov 2, 2019, 5:08 PM Tomas Rokicki <rokicki@gmail.com> wrote:
10, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9, 1 10, hit.
On Sat, Nov 2, 2019 at 1:59 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am pretty sure that nothing with a zero as the second digit can be a hit.
On Sat, Nov 2, 2019, 4:42 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-Fun,
We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process.
If, at some stage, the concatenation starts with the string K, we have a hit.
K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29
Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34
Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,...
HIT!
Are all integers potential hits? Best É.
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My own oops. I meant to ask how 110 begins with 10, but I think Tom received my intended message anyway. On Sat, Nov 2, 2019, 5:15 PM Tomas Rokicki <rokicki@gmail.com> wrote:
Oops.
On Sat, Nov 2, 2019 at 2:14 PM Allan Wechsler <acwacw@gmail.com> wrote:
How does 110 begin with "11"?
On Sat, Nov 2, 2019, 5:08 PM Tomas Rokicki <rokicki@gmail.com> wrote:
10, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9, 1 10, hit.
On Sat, Nov 2, 2019 at 1:59 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am pretty sure that nothing with a zero as the second digit can be a hit.
On Sat, Nov 2, 2019, 4:42 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-Fun,
We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process.
If, at some stage, the concatenation starts with the string K, we have a hit.
K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29
Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34
Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,...
HIT!
Are all integers potential hits? Best É.
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EA: "Are all integers potential hits?" No, only a few. Because a hit involves only as many digits as one's starting number (call it d), after an accumulation and concatenation just drop any digits after the first d. This shortcut creates a finite space and numbers traversing it will necessarily loop. But the loop, i.e. the eventually repeated term, is not likely the first term but rather some number subsequent to it. There are 29 two-digit integers that hit. Perhaps surprisingly, each of these may be evolved into a three-digit integer that hits by appending one specific digit. Each of these in turn may be evolved into a four-digit integer that hits by (again) appending a specific digit. And so on. If that holds true, then these 29 100-digit numbers http://chesswanks.com/num/twenty-nine.txt encapsulate all 2-to-100-digit solutions simply by reading the prescribed number of digits left to right.
Many thanks Hans,
If that holds true,
... I'm quite sure this is true -- but I cannot prove it :/ This is anyway how I found my example. And the rest of your mail is very interesting, thanks! Best, É.
Le 3 nov. 2019 à 01:20, Hans Havermann <gladhobo@bell.net> a écrit :
If that holds true,
I didn't study this question, but is there a sequence here? The "hits" sequence? Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Nov 2, 2019 at 8:32 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Many thanks Hans,
If that holds true,
... I'm quite sure this is true -- but I cannot prove it :/ This is anyway how I found my example. And the rest of your mail is very interesting, thanks! Best, É.
Le 3 nov. 2019 à 01:20, Hans Havermann <gladhobo@bell.net> a écrit :
If that holds true,
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HH: "... these 29 100-digit numbers < http://chesswanks.com/num/twenty-nine.txt > encapsulate all 2-to-100-digit solutions simply by reading the prescribed number of digits left to right." In Mathematica: In[1]:= ace[n_]:=(d=IntegerDigits[n];FromDigits[Take[Flatten[IntegerDigits[Accumulate[d]]],Length[d]]]) In[2]:= q={11,12,13,14,15,16,17,18,19,21,23,25,27,29,31,34,37,41,45,49,51,56,61,67,71,78,81,89,91}; In[3]:= Table[Length[NestWhileList[ace,q[[i]],Unequal,All]]-1,{i,29}] Out[3]:= {9,9,9,9,9,9,9,9,9,5,5,5,5,5,3,3,3,3,3,3,2,2,2,2,2,2,2,2,1} The output is the number of steps required to get each of the 29 two-digit solutions back to itself. Nine steps for the nine numbers starting with 1; five steps for the five numbers starting with 2; three steps for each of the three numbers starting with 3 or 4; two steps for each of the two numbers starting with 5, 6, 7, or 8; and one step for the one number starting with 9. The number of steps doesn't change as we increase the size of our starting integer. The first digit alone determines the number of steps. In[4]:= NestWhileList[ace,21112121319202729333,Unequal,All]//TableForm Out[4]:= 21112121319202729333 23457810111415242626 25914212930303132333 27161721232426353838 29101617242627293234 21112121319202729333 Here the five steps illustrate that the cycle reproduces the other solutions starting, in this case, with 2. This is true for all 29 solutions regardless of digit length. Finally, I thought that the solution starting with 9 (since it immediately produces itself) was special. Let's have a look: 910101111121314151618192223272833344041495... Do you see it? In[5]:= x=IntegerDigits[910101111121314151618192223272833344041495]; Accumulate [x] Out[5]:= {9, 10, 10, 11, 11, 12, 13, 14, 15, 16, 18, 19, 22, 23, 27, 28, 33, 34, 40, 41, 49, 50, 59, 61, 63, 65, 68, 70, 77, 79, 87, 90, 93, 96, 100, 104, 104, 108, 109, 113, 122, 127} It's already in the OEIS: https://oeis.org/A240919
participants (5)
-
Allan Wechsler -
Hans Havermann -
Neil Sloane -
Tomas Rokicki -
Éric Angelini