Now, as I have said, our puzzle may be solved at once by merely converting 1,000,000 dollars to the septenary scale.  Keep on dividing this number by 7 until there is nothing more left to divide, and the remainders will be found to be 11333311 which is 1,000,000 expressed in the septenary scale.  Therefore, 1 gift of 1 dollar, 1 gift of 7 dollars, 3 gifts of 49 dollars, 3 gifts of 343 dollars, 3 gifts of 2,401 dollars, 3 gifts of 16,807 dollars, 1 gift of 117,649 dollars, and one substantial gift of 823,543 dollars, satisfactorily solves our problem.  And it is the only possible solution.  It is thus seen that no “trials” are necessary; by converting to the septenary scale of notation we go direct to the answer.

17.—­THE PUZZLING MONEY BOXES.

The correct answer to this puzzle is as follows:  John put into his money-box two double florins (8s.), William a half-sovereign and a florin (12s.), Charles a crown (5s.), and Thomas a sovereign (20s.).  There are six coins in all, of a total value of 45s.  If John had 2s. more, William 2s. less, Charles twice as much, and Thomas half as much as they really possessed, they would each have had exactly 10s.

18.—­THE MARKET WOMEN.

The price received was in every case 105 farthings.  Therefore the greatest number of women is eight, as the goods could only be sold at the following rates:  105 lbs. at 1 farthing, 35 at 3, 21 at 5, 15 at 7, 7 at 15, 5 at 21, 3 at 35, and 1 lb. at 105 farthings.

19.—­THE NEW YEAR’S EVE SUPPERS.

The company present on the occasion must have consisted of seven pairs, ten single men, and one single lady.  Thus, there were twenty-five persons in all, and at the prices stated they would pay exactly L5 together.

20.—­BEEF AND SAUSAGES.

The lady bought 48 lbs. of beef at 2s., and the same quantity of sausages at 1s. 6d., thus spending L8, 8s.  Had she bought 42 lbs. of beef and 56 lbs. of sausages she would have spent L4, 4s. on each, and have obtained 98 lbs. instead of 96 lbs.—­a gain in weight of 2 lbs.

21.—­A DEAL IN APPLES.

I was first offered sixteen apples for my shilling, which would be at the rate of ninepence a dozen.  The two extra apples gave me eighteen for a shilling, which is at the rate of eightpence a dozen, or one penny a dozen less than the first price asked.

22.—­A DEAL IN EGGS.

The man must have bought ten eggs at fivepence, ten eggs at one penny, and eighty eggs at a halfpenny.  He would then have one hundred eggs at a cost of eight shillings and fourpence, and the same number of eggs of two of the qualities.

23.—­THE CHRISTMAS-BOXES.

The distribution took place “some years ago,” when the fourpenny-piece was in circulation.  Nineteen persons must each have received nineteen pence.  There are five different ways in which this sum may have been paid in silver coins.  We need only use two of these ways.  Thus if fourteen men each received four four-penny-pieces and one threepenny-piece, and five men each received five threepenny-pieces and one fourpenny-piece, each man would receive nineteen pence, and there would be exactly one hundred coins of a total value of L1, 10s. 1d.