Solutions to Puzzles of the Week

1. The digits 0, 5, 7, and 9 do not appear because multiplication of two elements of {1 2 3 4 6 8} can, with one exception, never yield 0, 5, 7, or 9. The one exception is 3*3, but you'll never have two 3's in a row to yield a 9. Given the starting configuration 12, you always append either a single even digit or an odd digit followed by an even digit. Thus you can never have two odd digits in a row, so you'll never get 3*3=9.

You get arbitrarily long sequences of 8's because a sequence of three or more 8's begets a longer sequence of 8's later on. The given sequence shows an early subsequence of three 8's. In general, 888... (n digits) --> 646464... (2n-2 digits) --> 242424... (4n-6 digits) --> 888... (4n-7 digits).

2.

3. SSS-E-N-WWW-S-E-N-W-NNN-E-S-W-SS-EEEE-NN-W-N-WW-S-W-SS-EEE-NN-W-N-WW-SSS-E-N-EE-S-W-NN
4.
5.
6.

7.

8.

9. Here is one possible solution:

10. 891, 1170, 1290, 2931, 51070, 147970, 75914061
11. W-S-EEE-NNNN-W-SS-E-NN-W-S-W-S-E-NN-W-NN-E-S-E-SS-WWW-NN-E-N-E-SS
12. N-W-SSS-EE-N-E-N-E-S-WW-S-E-NNN-WW-S-E-S-EE-N-WW-S-EE-NN-W-S-W-S-E-N-WW-N-EE-S-W-S-S
13. LLDLURRRDRULLLDLURRD (lul) ULLDRUR (uru) LDLURU (ll) DLDRUL (dlu) RRRRUULLLDDLURU (rrr) DRDRUUL (uuu) RDDLULLDRUURRRDDDLURUU (llll) DDLDRUUULLLDDLDRRURUULLLU (lll) DRR (ddd) RDDLULU (lll) UL (dd) RDDLDRUURRRDDLLLL (dru) RRRRUULLLDDLURU (rr) DRDRUUL (urd) RDDLULU (lll) DLDRUURRRDDLLLU (drru)
14. Possible solutions are:
EFSST
IOIEH
GUXVR
HR.EE
TWONE
TWELVE
HS.TLE
GEEEER
IVVNUH
EEIOWT
NNFXIS

15. MAINE, new hampshire, vermont, massachussets, rhode island, connecticut, new york, new jersey, delaware, pennsylvania, ohio, michigan, indiana, kentucky, west virginia, MARYLAND, virginia, north carolina, south carolina, georgia, florida, alabama, tennessee, missouri, illinois, wisconsin, minnesota, north dakota, montana, idaho, WASHINGTON, oregon, nevada, utah, wyoming, south dakota, iowa, nebraska, KANSAS, colorado, okhaloma, arkansas, mississippi, lousianna, texas, new mexico, arizona, CALIFORNIA.
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17.
18.

19. Matthew Daly found that two states suffice: visit Tennessee 6th and Ohio 18th!

20.
21.
22.
23.
A = 214 = 194481 B = 244 = 331776 C = 5824 114733948176
24.
25.
1108033241108033241 1220096161220096161 4312080784312080784
4432132964432132964 4880384644880384644 9024307889024307889
9972299169972299169
26.
27.
28.
29.
30. Arrange the cubes in two layers, rotated with respect to each other, as shown. The middle cubes in each layer should be pulled out a little so they only contact cubes in their layer.

31.
32.
191520
102483
165186
274212
33. 301401, 242064, 123201
34. September 26, 2003.
35.
36. 721 + 712 + 271 + 172 + 127 = 2003
37.
567
432
38.
39. A=21, B=26, C=64, D=37, E=62
40. 200:3 → 400:0:9 → 1:600:81 → 3:665:6:2 → 442:2:74 → 2:0:0:8:44 → 2004.
Keith Lynch found : 200:3 → 40009 → 1:600:7:2008:1 → 4:39:21:1:5 → 2004.
41. 222/2 - 2 × 22 = 2004
42. 19576 and 77714446555
43.
44.
45.
46.
47.
48.
49. Their centers of gravity are the centers of one of the shapes.
50. The minimum side length is a little more than 1.961.

51. (57 + 70) / 39 = 2005
52.
53. (8 - 7×6) × (5 - 43) = 2006
54.
55. The digits are 2 and 8.
56.
57. {2,2,2} → {2,1,2,2,2} → {1,1,2,2} → {1,1,2,1,2,2} → {1,1,1,1,2} → {1,2,1,2} → {1,1,1} → {2,1} → {1,2,2,1} → {1,2,1,2,2,1} → {1,1,1,2,1} → {2,1,2,1} → {1,2,2,1,2,1} → {1,2,1,1,1}
58.
59. 2469
60.
3 × (674 - 5) = 2007 (4 + 5) × (63 + 7) = 2007
61.
62.
2 + 2^(.2) = 3.149 (7.7)^(.7 x .7) = 2.7188 8^(-8^(-.8 x .8)) = .57723
63.
8 × (3 × 81 + 8) = 2008 8 × 9 × 28 - 8 = 2008
8 × (44 - 4 - 1) = 2008 3 × 8 × 84 - 8 = 2008
8 × (28 - 8 + 3) = 2008 8 × (44 - 9 + 4) = 2008
2 × (999 + 5) = 2008 8 × (4 × 8 × 8 - 5) = 2008
7 × 288 - 8 = 2008 6 × 6 × 56 - 8 = 2008
64.
65.
25 + 26 + 27 + 45 + 63 = 52 + 62 + 72 + 54 + 36
and
23 + 27 + 36 + 54 + 82 = 32 + 72 + 63 + 45 + 28
66.
67.
68. (( 7 × 7) - 7) × 7 - 7) × 7 = 2009
69.
70.
71.