268 



• KNOV/LEDGS • 



(May 4, 1881 



sliino down tlio tube; and (2), if it bociimo less prrpcndicular, in- 

 stead of ehininp down the tube on Dec. 21, it would do so on two 

 days, one before and the other after the winter solstice. 



5. The idea of a chamber diirk all the year, except on the one day 

 when the snn is in an exceptional position, is thoroughly Egyptian ; 

 e.(j.^ in one of the rock temples of the sun its rays only roach the 

 inner sanctuary at sunrise on the longest day. 



6. If this guess as to use of tube F. G. is correct, it would 

 account — 



(1) For the accurate orientation of Pyramid,'without which the 

 tube could not be constructed. 



(2) For one use of the grand (transit) gallery, which would be 

 needed for observations as to the exact elevation of noonday sun at 

 winter solstice. 



(3) For the Pyramid being closed on completion. The secret of 

 the entrance and the tube being kept by tradition among the priestly 

 caste, so that at the end of, say 500 and 1,000 years, by which time 

 the change (if any) could bo observed, it could be re-entered. 



7. Possibly yon may have measurements of sufficient accuracy to 

 decide if the direction of the tube is such as to make this supposed 

 nse jiossible. or, if it has not been observed, by mentioning the 

 matter in Knowledge, some future observer could settle the point. 



A. M. Deaxe. 

 P.S. — I was amused,to see you printed my symmetrical solution 

 of the twenty-one girls' problem in preference to the author's, but 

 I sh luld like to have seen the latter. 



[805] 

 divided 

 bored 1 

 so that 

 the 3rd 



Make 

 the 3rd, 



TWENTY-ONE SCHOOL-GIEL PUZZLE. 



— Let the girls bo designated by numbers, 1 to 21, and 

 into sets, each of tliree consecutive numbers, the sets num- 

 to 7. The numbers are considered as arranged in a circle, 

 we may count forward from 7 to 1, and say that set 3 is 

 set from set 7. 



two squares, one of the 1st. 2nd, and 1th sots, the other of 



5th, and 6th, as follows : — 



(Observe the order of writing the numbers in the second square.) 



These squares are divided vertically into six groups of three, 

 which, with the seventh set — 19, 20, 21 — as a seventh group, makes 

 an arrangement of girls for the first day. Make the second day's 

 arrangement from the first, by adding 3 to each number and 

 subtracting 21, when the addition makes more than 21 ; the third 

 day's arrangement from the second's in like manner ; and so on 

 until you have arrangements for seven days. Then take the sets 

 composing one of the above squares, and make a third square of 

 them in this wav : — 



The three columns of this square furnish one gi-oup for the ar- 

 rangement for each of the three remaining days, and the other 

 groups arc formed from them by successive additions of 3, iSrc, 

 just as the arrangements for the days from the second to the 

 seventh were formed each from the arrangement of the preceding 

 day. 



Proof. — By the successive additions of 3. the three squares given 

 above may be considered as extended into tliioe sets of squares, and 

 in each set of scpiares each line is occujiied by all the sets of 

 tliree consecutive numbers successively. In every square two con- 

 secutive sets are combined, two sets whoso numbers differ by 1 and 

 two whoso numbers differ by 2, so that in each set of squares each 

 set of numbers is combined with every other set. Htit, when the 

 same two sets of numbers are combined in different squares, they 

 are combined in different ways. For instance, the first and second 

 sets are combined in the first sot of squares — 1 with 4, 2 with 5, 

 3 with G; in tho second set— 1 with 5, 2 with 0, 3 with 4 ; and in 

 the third set— 1 with 6, 2 with 4, 3 with 5 ; so we see that not only 

 is each set combined with all tho other sets, but every number of 

 every set is combined in a gron]) of three with every number of 

 every other set, and, a.s on each of tho first seven days one set 

 forms a group by itself, every oni- of the 21 numbers is combined 

 with each of the other numbers. Ai.oekxox Bkav. 



"31" GAilE. 

 [806] — I think yon might, perhaps, be interested by a game or 

 puzzle of the above name, which has lately made its aj>pcaranco in 

 New York, and of which the following sketch may give yon an 

 idea : — 



The sketch represents a box divided by longitudinal partitions 

 into six parts, and containing twenty-four numbered blocks. The 

 blocks being first placed on one side of the bo.\, two players alter- 

 nately move one block at a time to the other side, keeping count of 

 the sum of the numbers on the blocks moved, as if they were 

 playing cribbage. The player who makes 31, or who comes so near 

 31 that the other cannot play without making more than 31, wins 

 the game. 



The inventor of the game seems to have intended that the blocks 

 should always be placed in the box in the above order. Even with 

 this restriction, the game presents a tolerable problem, though it 

 has not much variety. But if the blocks might be placed in any 

 way you pleased, four in each row, there would be a great variety 

 of problems, the number of essentially different arrangements 

 being, I believe, 4,509,264,631,875. The problems wonld often be 

 exceedingly complicated, and there would be room for a good deal 

 of skill in play. 



The box of blocks is by no means necessary in order to play the 

 game. You can readily see how it could be played with cards, and, 

 fiom the above regular arrangement, with a pencil and paper. 



Algebxos Bkav. 



THE SQUIEREL PUZZLE. 



[807] — As a reader of Kxowxedge, I hope you will excuse what 

 I am about to put before you, but I cannot arrive at the same con- 

 clusions as you in regard to the squirrel and the man going roimd 

 him. 



Supposing the squirrel that sits on the post holds apiece of string 

 in his mouth and the man has hold of the other end and walks 

 round the post, the squirrel turning at the same time ; if your con- 

 clusions are correct, the man must go around the piece of string 

 the squirrel holds in his mouth, and, consequently, round the piece 

 the man holds, or round himself [he does rotate — R. P.] ; otherwise, 

 there must be a point in the string where the man leaves off going 

 round it. I cannot see how the man goes round the string or 

 squirrel in the same sense that he has gone ronnd the post. 



You might as well say that we go round the earth every time the 

 earth turns on her axis. ^^^ ^® ^^^ •' — ^- P*] 



Perhaps, after all, wo only disagree as to what is going round au 

 object. I consider that in going ronnd an object we see during 

 the revolution every side of it, or that every side is presented to ns 

 in its turn. W. Smith. 



[In what way does the expression going round an object imply 

 seeing every side of it. Suppose the man sliut his eyes, would that 

 make any difference ? Or, suppose the man stood still and the 

 sciuirrel turned round, so as to show him every side — would the 

 stationary man have gone round the squirrel ? In the original sup- 

 posed case of the squirrel turning, he simply rotated ; the string 

 shares the motion of revolution. We cannot deal with that ca."*" 

 except by using the more rigid definition of revolution. — R. P.] 



MUSICAL QUESTION. 



[SOS] —How is it that the same chord, if struck in two different 

 keys (say C Major and C-sharp Major), has a different sound — 

 irrespective of pitch — when a pianoforte is tuned to equal tem- 

 l>erainent ? 



I have heard that there is a solution, but I have never comet 

 across it, and many musical friends are unable to enlighten me onj 

 the subject. I therefore appeal to you, Mr. Editor. I 



Walter Jones, i 



[The question is left to musical correspondents. It has al.vayj^ 

 seemed to me that there is something akin in the fact mentioned liy, 

 Mr. Jones to the way in which a lumincuis surface presents » d'fj 



