perplexus.info :: Numbers : Digit Replacement and Square Settlement

Digit Replacement and Square Settlement (Posted on 2022-04-11)

Consider a perfect square N having 1 as the first digit (reading from left).
Determine the minimum value of N such that it remains a perfect square when 1 is replaced by 2.
Find, if possible, the next higher value of N less than 10¹⁰ with this property.
Otherwise, prove its non-existence.
Note: Computer-program based methodology apart from semi-analytic solution is permissible.

No Solution Yet Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 1 of 5

11025=105^2

21025=145^2

1102500=1050^2

2102500=1450^2

By noting b^2-a^2= 10^n and checking the solutions given by wolfram|alpha to see if any give the right size.

addendum:

next solution

11390625=3375^2

21390625=4625^2

(these come from 1250*8000=10^7 being just the right size)

Edited on April 11, 2022, 7:55 am

Posted by Jer on 2022-04-11 07:39:00

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