February 1, 1887.] 



♦ KNOWLEDGE ♦ 



83 



force, being associated with a ray of light. Therefore, what- 

 ever special virtues may belong to that class of i-ays called 

 the " chemical," those special \nrtues must con.sist in the 

 mechanical action of the waves of this class upon certain 

 substances. 



[I sympathise much with Mr. Thomas's purpose in the 

 above communication. But I fear that the strict accuracy 

 for which he stickles can hardly be secured in company with 

 gieat conciseness. Let anyone try to express concisely the 

 idea conveyed, though not with strictly verbal accuracy, 

 by — for example — the statement that the light of such and 

 such a star took ten years to travel across interstellar space 

 to the earth. — E. P.] 



OUR PUZZLES. 



OME correspondents complain that we give 

 too much room to puzzles, and they are too 

 ditficult. Our puzzles have mostly been 

 intended as mathematical recreations. Here 

 are three which are very easy — perhaps 

 familiar to most readers — but interesting 

 as studies. 



I'lzzLE XIX. Shoio hoic to cut a regular tetahedron 

 (e.qiiilaferril triangular pyramid) so that the face, cut shall 

 be a square: also shoio how to plug a square hole with a. 

 tetahedron. 



PczzLE XX. Shoic hoic to cut a cube so that the cut face 

 shall he a regular hixagoii : also show how to Jjlug a hexagonal 

 hole with a cube. 



Puzzle XXI. Shou- hoio to cut a regular octahedron (a 

 double square pyramvl with triangular faces) so that the cut 

 face sltall he a reijular hexagon ; and show how to plug a 

 hexagonal hole with an octahedron. 



The dodecahedron and icosahedron can be cut with 

 decagonal faces or used to fill decagonal holes. The former 

 also served the editor as a basis for his library and school 

 star atlases, and for the equal-surface atlas whose maps are 

 now in progiess. 



SOLUTION OF LAST MONTH'S PUZZLES. 



■ UZZLE XYI. Box No. 1 is 11 inches square, 

 inside measurement, in the ba.se; and ll/^y 

 inches deep ; and the fruiterer desires to pack 

 in it 200 oranges. 



He can effect this by the arrangement illus- 

 trated in fig. 1, a, where the '25 darker circles 

 represent the lowest layer of oranges, and the 

 '2o hght circles the layer next above it, the successive layers 

 corresponding alternately with these, so that the dark circles 

 may be regarded as showing the positions of the oranges in 

 the 1st, .3rd, 5th, and 7th layers, the light circles represent- 

 ing the positions in the 2nd, 4th, 6th, and 8th. 



To determine the height of the centres in any layer above 

 those in the next lower Layer, let a plane be supposed taken 

 through the centres of the oranges a, b in the lowest layer 

 and c in the layer above. (This plane will be vertical if the 

 layers are horizontal.) Then we have the section shown in 

 fig. 1, b, where the centres a, b, c are represented by 

 A, R, c. Since ac^cb=:2 inches, CM=v/2 = l-41i inch. 

 Thus each layer rises r414 inch above the last; and since 

 the lowest layer rises 2 inches above the bottom of the box, 

 leaving 9^^ inches to the top, we are limited to as many 

 layers above the bottom one as the (whole) number of times 

 1414 inch is contained in O,',, inches; i.e. there are 

 7 layers above the lowest (since 1-414 x 7=9-898). Thus 



there can be 8 layers, and it needs no proof that there can 

 be 25 in each layer, arranged as in the figure. Consequently 

 200 oranges can be packed in this way. It will be shown 



Rftt.b. 



presently that by no other arrangement can more than 

 200 oranges be packed in Box No. 1. 



Puzzle XVII. Box No. 2 has a base 12 inches by 

 11^ inches, inside measurement, and is llf*,, inches deep. 

 The fruiterer has to pack in it 2.31 oranges. 



He can effect this liy the arrangement shown in fig. 2, o, 

 where the dark and light circles are to be understood as 

 before. We get in G and 5 in alternate rows, and 6 rows 

 fall easily within the Hi inches in the lowest row, for we 

 see from fig. 2, b, that h and c, the centres of the 2nd row, 

 fiiU further from the side of the box than a does, by the 

 distance AM = ,y 3, or 1-73 inch. Hence, since the 1st row 

 reaches 2 inches from the side, the sixth reaches 2-1-5 x 1-73 

 = 10'65. But it is further manifest that e, the centre of an 



orange of 2nd laj-er, lies farther than a from the side by a 

 distance equal to i ao, or om in fig. 2, b. That is each row 

 in the upper layer overlaps by ^ y/'3, or -58 inch, a row in 

 the lower layer. (This, of course, is true of every layer ; 

 only in the aiTangement illustrated the rows in the lower 

 layer overlap the rows of the layer above towards, not from, 

 the side shown at the top of the figiire. Adding -58 to 

 10-65, we get 11-23 inches, so that the upper layer, like the 

 lower, can contain 6 rows ; and yet lie within the breadth 

 (11:^ inches) of the base. Thus in each layer we get 6 



