The problem

Show that the sequence, $49, 4489, 444889, \dots$, gotten by inserting the digits $48$ in the middle of the previous number (all in base $10$), consists only of perfect squares.

has become a classic. For some reason, I got curious as to who actually discovered this.

After doing some research, seems like this problem was used in the Stanford University Competitive Exam in Mathematics for high school seniors, in the year 1964. One pdf which has this is here: http://www.computing-wisdom.com/jstor/stanford-exam.pdf. It appears as problem 64.2.

The pdf also mentions that there is a booklet which gives references to previous appearances of the problems from the above exam, but that seemed to be a dead-end regarding this particular problem.

Does anyone know who originally discovered this little gem? More interestingly, is it known how this was discovered?

Update 1

On some more research I found this book from 1903: Algebra Part II by E.M. Langley and S.R.N. Bradly which has this as an exercise on page 180. The question seems to have been phrased in such a way as to claim ownership and also tells how it was discovered. I guess we just need confirmation now.

Update 2

More digging reveals this German book by Dr. Hermann Schubert, Mathematische Mussestunden, which has this on page 24. The book was published in 1900, but the preface seems to be dated earlier. If someone can read German, perhaps maybe they can read and see what the book claims about the origins of this problem. The book seems to have a list of references at the beginning.

  • $\begingroup$ Not sure of the tags, please feel free to change/add. $\endgroup$ – Aryabhata Jan 10 '11 at 4:02
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    $\begingroup$ Yes, Dickson's History of the Theory of Numbers, volume 1, Chapter XX ("Properties of the Digits of Numbers") is a place to look. I just glanced through it, though, and didn't find specifically your phenomenon, but there is enough there to convince me that this was well known prior to 1900, so I think you'll have to dig to find the earliest source. Dickson mentions many square related digit results, and also mentions "similar problems" and "collections of problems". $\endgroup$ – Matthew Conroy Jan 10 '11 at 18:32
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    $\begingroup$ @Matt: I found a book from 1903 that has this, and which might be the source (see update to question). But, I agree with you and am guessing this has got to be older. $\endgroup$ – Aryabhata Jan 10 '11 at 18:38
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    $\begingroup$ I think my rudimentary German is enough for me to claim that the Schubert book makes no claims as to the origin of the problem, at least not in the vicinity of the problem itself. This is the kind of phenomenon that has been discovered over and over again since the invention of the decimal system, and I'm sure numerous folks have come up with myriad (most virtually identical) independent proofs. As a result, I expect most authors would not give a reference for it in their works. $\endgroup$ – Matthew Conroy Jan 11 '11 at 1:54
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    $\begingroup$ @Matt: Yeah, I agree. I was just wondering if that book made any references. I found this from the references of a 1912 book, so some do give references. But, I guess you are right, this problem is likely to be very old and finding the oldest reference is probably the best we can do. $\endgroup$ – Aryabhata Jan 11 '11 at 2:05


This fact appeared as a problem in the October 1, 1889 issue of the Journal de Mathématiques Elémentaires, page 160. The problem is attributed to F. Briganti of the Ecole de industrielle de Fermo.

I don't yet have the rep to leave a comment so I'll leave this as an answer...

This fact appears in a note of M. C.-A. Laisant in 1892. See here, page 77. He remarks that this "cette remarque, paraît-il, a été faite depuis longtemps" (this remark, it seems, was made a long time ago). No references are given. Later in the same volume, in the minutes of the February 27, 1892 meeting of the Société Philomathique de Paris, this fact is referred to as a "fait connu" - a known fact - so perhaps this was already known at the time.

So it seems that you will have to dig much farther back to find the original source.

(The page numbers in this Google Books version don't seem to work, but just search for 4489 and you should find the two references to this fact.)

  • $\begingroup$ @Aryabhata - I added a new source from 1889. My guess is that you won't find anything much older on the internet. $\endgroup$ – Corey Jun 22 '11 at 21:05
  • $\begingroup$ Thank you very much! I will wait a few more days (bounty period) and accept this as the answer (and the bounty answer). (Just in case F.Briganti got it from elsewhere!) $\endgroup$ – Aryabhata Jun 22 '11 at 21:05

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