Ken's POTW

Books and Cannonballs
Two more from Sam Loyd, then I'll put my book back on its shelf:
1. You have nine large books, labeled 1,2,3,4,5,6,7,8,9 on a bookshelf with two shelves. Placing volumes 6,7,2,9 on the top shelf and 1,3,4,5,8 on the bottom shelf, you create a fraction (6729/13458) exactly equal to 1/2. Is it possible to find other arrangements, using all nine volumes, that will make fractions equivalent to 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, and 1/9? [Tricks like inverting the books aren't needed, but I would be happy to post any solutions found in this way.]

2. James is an adept cannonball stacker. In each of the cases below, he is asked to split one pyramid of cannonballs into two smaller ones. Assuming James can complete each request, what are the minimum (and perhaps other representative) sizes of each pyramid?
• Split a pyramid with a square base into two smaller pyramids, each with a square base.
• Split a pyramid with a square base into two smaller pyramids, each with a triangular base.
• Split a pyramid with a triangular base into two smaller pyramids, each with a triangular base. [A simple answer can be found if the smaller pyramids are identical. Try to also find an answer with them unequal.]
• Split a pyramid with a triangular base into two smaller pyramids, each with a square base.

Source: Mathematical Puzzles of Sam Loyd, Volume Two, edited by Martin Gardner, Dover Publications, New York, 1960, #74, #78.

Solution:
1. Philippe Fondanaiche (Paris, France) found several solutions. Here is his sampling:
```There are respectively:
12 fractions equal to 1/2: 6729/13458,  6792/13584, etc....
2 fractions equal to 1/3: 5823/17469, 5832/17496
4 fractions equal to 1/4: 3942/15768, 4392/17568, etc...
12 fractions equal to 1/5: 2697/13485, 2769/13845, etc...
3 fractions equal to 1/6: 2943/17658, 4653/17918, etc...
7 fractions equal to 1/7: 2394/16758, 2638/18459, etc...
46 fractions equal to 1/8: 3187/25496, 4589/36712 etc...
3 fractions equal to 1/9: 6381/57429, 6471/58239, etc...  ```
I asked him to send all of his solutions. They are included below.

2. Philippe Fondanaiche and Kirk Bresniker each sent solutions to problem 2. Kirk Bresniker found all solutions for side-lengths less than 100:
```square -> 2 squares
116795 S(70) = 4900 S(24) + 111895 S(69)
56980 S(55) = 25585 S(42) + 31395 S(45)

square -> 2 triangles
S(i+1) = T(i) + T(i+1), plus:
140 S(7) = 20 T(4) + 120 T(8)
3311 S(21) = 35 T(5) + 3276 T(26)
1785 S(17) = 455 T(13) + 1330 T(19)
10416 S(31) = 3276 T(26) + 7140 T(34)
40425 S(49) = 14190 T(43) + 26235 T(53)
121836 S(71) = 45760 T(64) + 76076 T(76)

triangle -> 2 squares
56 T(6) = 1 S(1) + 55 S(5)
286 T(11) = 1 S(1) + 285 S(9)
10 T(3) = 5 S(2) + 5 S(2)
35 T(5) = 5 S(2) + 30 S(4)
680 T(15) = 30 S(4) + 650 S(12)
2925 T(25) = 55 S(5) + 2870 S(20)
20825 T(49) = 285 S(9) + 20540 S(39)
26235 T(53) = 650 S(12) + 25585 S(42)
7770 T(35) = 2870 S(20) + 4900 S(24)
152096 T(96) = 2870 S(20) + 149226 S(76)
125580 T(90) = 13685 S(34) + 111895 S(69)

triangle -> 2 triangles
20 T(4) = 10 T(3) + 10 T(3)
680 T(15) = 120 T(8) + 560 T(14)
29260 T(55) = 1540 T(20) + 27720 T(54)
34220 T(58) = 4960 T(30) + 29260 T(55)
70300 T(74) = 10660 T(39) + 59640 T(70) ```
Philippe Fondanaiche's complete list of solution for the books:
```A       B       B/A

6381    57429   9
6471    58239   9
8361    75249   9

3187    25496   8
4589    36712   8
4591    36728   8
4689    37512   8
4691    37528   8
4769    38152   8
5237    41896   8
5371    42968   8
5789    46312   8
5791    46328   8
5839    46712   8
5892    47136   8
5916    47328   8
5921    47368   8
6479    51832   8
6741    53928   8
6789    54312   8
6791    54328   8
6839    54712   8
7123    56984   8
7312    58496   8
7364    58912   8
7416    59328   8
7421    59368   8
7894    63152   8
7941    63528   8
8174    65392   8
8179    65432   8
8394    67152   8
8419    67352   8
8439    67512   8
8932    71456   8
8942    71536   8
8953    71624   8
8954    71632   8
9156    73248   8
9158    73264   8
9182    73456   8
9316    74528   8
9321    74568   8
9352    74816   8
9416    75328   8
9421    75368   8
9523    76184   8
9531    76248   8
9541    76328   8

2394    16758   7
2637    18459   7
4527    31689   7
5274    36918   7
5418    37926   7
5976    41832   7
7614    53298   7

2943    17658   6
4653    27918   6
5697    34182   6

2697    13485   5
2769    13845   5
2937    14685   5
2967    14835   5
2973    14865   5
3297    16485   5
3729    18645   5
6297    31485   5
7629    38145   5
9237    46185   5
9627    48135   5
9723    48615   5

3942    15768   4
4392    17568   4
5796    23184   4
7956    31824   4

5823    17469   3
5832    17496   3

6729    13458   2
6792    13584   2
6927    13854   2
7269    14538   2
7293    14586   2
7329    14658   2
7692    15384   2
7923    15846   2
7932    15864   2
9267    18534   2
9273    18546   2
9327    18654   2
```

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