The Daily Herald

Squares.	Roots.	Squares.	Roots
139,854,276	11,826	587,432,169	24,237
152,843,769	12,363	589,324,176	24,276
157,326,849	12,543	597,362,481	24,441
215,384,976	14,676	615,387,249	24,807
245,893,761	15,681	627,953,481	25,059
254,817,369	15,963	653,927,184	25,572
326,597,184	18,072	672,935,481	25,941
361,874,529	19,023	697,435,281	26,409
375,468,129	19,377	714,653,289	26,733
382,945,761	19,569	735,982,641	27,129
385,297,641	19,629	743,816,529	27,273
529,874,361	23,019	842,973,156	29,034
537,219,684	23,178	847,159,236	29,106
549,386,721	23,439	923,187,456	30,384

Messrs. Editors:-

Having answered the mathematical question so modestly proposed in the Stamford Sentinel, by arranging the nine digits so that the square root can be extracted without a remainder, we thought it but fair in our turn to ask several additional questions relative to the same problem: All these (as we learn from the same paper,) the mathematician, for some unaccountable reason, declines answering, except one. Upon our intimation that several solutions may be obtained to the question, he ventures to answer this query by stating the number of solutions at nine, without, however, actually giving the solutions themselves, as would have been gratifying: Now instead of nine independent solutions, the question admits of at least twenty eight, viz. Roots and squares entered, as appears without distinction into the reply of the Stamford Sentinel to the article signed "Freshman;" but in the reply to ours, we are favored with another equally rare specimen of discrimination and accuracy. The author kindly undertakes to point out an error, which would certainly have remained undiscovered, had it not been for his singular penetration.-In the process, however, he contrives, by way of compensation probably, to run into four errors in as many lines. We stated that between the limits 123,456,789 and 987,654,321 there were about 20,000 squares. We will now give the exact number, viz. 20,315. But, says the Stamford Sentinel, "Mr. X. greatly mistakes when he puts down the number of squares the question will admit of at only about 20,000.-The nine digits admit of not less than 362,880 different combinations;-hence, as the question will admit of nine different answers, a person might "cut and try" 362,879 times before he would light upon either one of them." In the first place, it is almost too obvious for remark, that we do not put down the number of squares which the question will admit of at 20,000. What we say is, indeed, quite sufficient to show that the number of answers which belong to the question cannot be greater than about 20,000. In reality, however, this number is far less. Yet we greatly mistake, when (as is said) we put down the number at only about 20,000. And the proof of this is, that the nine digits admit of 362,880 different combinations. But it might perhaps be somewhat to the purpose, if we were also told how many of these combinations are squares. We would here remark that 362,880 should be substituted above in place of 362,780. The mistake of 7 for 8 might be thought a typographical error, had not repetition in a following line marked it as the blunder of the calculator. Finally, it is said, that as there are 362,880 different combinations of the nine digits, and 9 answers to the proposed question, a person might "cut and try" 362,879 times before one of these answers would be found. As to this we will only say, it is new to us that the difference between 9 and 362,880 is 362,879. But we have spent too much time in the correction of such a bundle of errors. We shall not, in addition, take the trouble at present, to state the mode by which we obtained our answers given above. The amusement of "cutting and trying" we are disposed to leave the Stamford Sentinel to pursue, without interference. In its article on the nine digits to which we have all along had reference, something is said about taking lessons in Q. E. D. We, in turn, would rather recommend that the writer of that article take lessons in A. B. C. or somewhere at a slight remove beyond; at least till he has learned to spell pole, a word which he uses twice, and each time writes it poll. In conclusion, we will barely remark, that we have given more attention to the problem than is suited either to its own importance, or the style in which it was proposed.- X.