9
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You are going to start from any number with distinct digits lower than four digits (such as $a$, $ab$, $abc$) and you will apply the rule below:

  • Take the square any number you want in the number (you may take the square the number itself too). (such as $1(02)$ -> $14$, or $12(3)$ -> $129$ or $(13)$ -> $169$)
  • Every number you found has to have distinct digits. (Fail example: $(12)3$-> $1443$ X)
  • You may apply this as many times as you want. ($(13)$ -> $169$ then $(16)9$->$2569$)

What is the biggest final number you can have?

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  • 1
    $\begingroup$ Biggest final number? or biggest starting number? $\endgroup$ – SteveV Sep 16 '18 at 16:33
  • $\begingroup$ is there a restriction about how many digits the number which we square can have? like, if I get a number like 1023 as an interim result (even though 1023 itself is not possible), am I allowed to square it all to get to 1046529? $\endgroup$ – elias Sep 16 '18 at 17:11
  • $\begingroup$ @elias "you may take the square the number itself too". written on the first rule. no restriction. $\endgroup$ – Oray Sep 16 '18 at 17:12
  • $\begingroup$ $1(69) -> 14761$. Isn't this a violation of the previous rule? $\endgroup$ – Wais Kamal Sep 16 '18 at 17:52
  • $\begingroup$ @WaisKamal ops true :) fixed it. $\endgroup$ – Oray Sep 16 '18 at 17:53
9
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Update:

I've found

9857240136 to be maximal (confirmed with programming).

Many possible ways to reach it, here is one starting with...

(4)2 -> 16(2) -> 1(6)4 -> 1(3)64 -> (1964) -> 3857(2)96 -> 3857(496) -> (3)85724016 -> 98572401(6) -> 9857240136

Previous attempt:

My new record is:

9810362574

which can be reached via

(5)74 -> (25)74 -> (625)74 -> 3(9)062574 -> (3)81062574 -> 9810(6)2574 -> 9810362574

First attempt:

Not proven to be maximal at all, but I managed to reach

10 digits with 4817093625

The steps are:

(705) -> 4(9)7025 -> 48170(25) -> 48170(6)25 -> 48170(3)625 -> 481709(6)25 -> 4817093625

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