The general solution to the problem is, in fact, this: 

      n
     C
      2n
    -----
    n + 1

where 2n equals the number of barrels.  The symbol C, of course, implies that we have to find how many combinations, or selections, we can make of 2n things, taken n at a time.

280.—­BUILDING THE TETRAHEDRON.

Take your constructed pyramid and hold it so that one stick only lies on the table.  Now, four sticks must branch off from it in different directions—­two at each end.  Any one of the five sticks may be left out of this connection; therefore the four may be selected in 5 different ways.  But these four matches may be placed in 24 different orders.  And as any match may be joined at either of its ends, they may further be varied (after their situations are settled for any particular arrangement) in 16 different ways.  In every arrangement the sixth stick may be added in 2 different ways.  Now multiply these results together, and we get 5 x 24 x 16 x 2 = 3,840 as the exact number of ways in which the pyramid may be constructed.  This method excludes all possibility of error.

A common cause of error is this.  If you calculate your combinations by working upwards from a basic triangle lying on the table, you will get half the correct number of ways, because you overlook the fact that an equal number of pyramids may be built on that triangle downwards, so to speak, through the table.  They are, in fact, reflections of the others, and examples from the two sets of pyramids cannot be set up to resemble one another—­except under fourth dimensional conditions!

281.—­PAINTING A PYRAMID.

It will be convenient to imagine that we are painting our pyramids on the flat cardboard, as in the diagrams, before folding up.  Now, if we take any four colours (say red, blue, green, and yellow), they may be applied in only 2 distinctive ways, as shown in Figs, 1 and 2.  Any other way will only result in one of these when the pyramids are folded up.  If we take any three colours, they may be applied in the 3 ways shown in Figs. 3, 4, and 5.  If we take any two colours, they may be applied in the 3 ways shown in Figs. 6, 7, and 8.  If we take any single colour, it may obviously be applied in only 1 way.  But four colours may be selected in 35 ways out of seven; three in 35 ways; two in 21 ways; and one colour in 7 ways.  Therefore 35 applied in 2 ways = 70; 35 in 3 ways = 105; 21 in 3 ways = 63; and 7 in 1 way = 7.  Consequently the pyramid may be painted in 245 different ways (70 + 105 + 63 + 7), using the seven colours of the solar spectrum in accordance with the conditions of the puzzle.

[Illustration: