Andrew, 7 years old, came from Idaho to work with Don March 11-16, 2002
Monday March 11:
1. Andrew worked on one quadratic equation x2 -5x + 6 = 0 and found the answers to be 2 and 3. He didn't want to do another one.
2. He did a fine job with 1/2 + 1/4 + 1/8. and they left it like that.
3. Andrew worked on the peg game and quickly did the 4 pegs on each side. He didn't want to try to get the rule for the number of moves to interchange the pegs.
4. He built the 3D multiplication tower and found patterns in it.
5. He worked with the balance scale.
6. Andrew spent about 20 minutes on the 15-puzzle (to make a magic square). He did not finish at this time.
7. He knew that 7! (7 factorial, 7*6*5*4*3*2*1) = 5040; then Don asked him what (7/8)! would be. He thought about that and Don showed him how to get this in Mathematica:
N[3!,6]
6.
N[7/8,100]
0.87500000000000000000000000000000000000000000000000000000
N[(7/8)!,100]
0.95344581274503483234582966071503134975438630929661948044949
10378 469231152826445654396409654816612141...
N[(7/8)!,300]
0.95344581274503483234582966071503134975438630929661948044949103784692311528264456543964096548166121405845216
7132874977509016301665921526845645694480970398194839713297211017842938214232897892125679332349482265192401378
390866277261354746253556573374115429536865238720877187299353572851138035873020603163...
N[(7/8)!,1000]
0.953445812745034832345829660715031349754386309296619480449491037846923115282644565439640965481661214058452167
13287497750901630166592152684564569448097039819483971329721101784293821423289789212567933234948226519240137839
08662772613547462535565733741154295368652387208771872993535728511380358730206031626126398751967637538748777745
70059274671099803902262762360237093118867011000617334054618989474203428979193231486287353541980901572944817657
725185245187274073034416102956768637080813555103298961166812621454317369849743740062122474070228381002150410545
265569934860522541531456577308441500335223864498311793030652073674901871727625625065613925200442445425131793421
870776821764883202915913134156382657740007230648180828011491722536811007413163740282873916511398029495119116710
548648726559960528940550613842565733862064638941300328070477753994636778246793198461581090334048257779182144052
284701414581136385738221573981232926975607038584964439216030048470648581947907226967923476682851280044748065940
7219623
N[(7/8)!
N[(6/8)!
N[(6/8)! ,1000]
1781910248722624942747207796094785326912949878670834394576675582382626760914987434369615695839579788695071752
8183978572036929125931985890480264517814268529574317494250567113331663274866446023016180566578205736442662767
7580916621434578278721074474648050668436772836353485403770006754469568560841859350120734223542597876690106384
6168804046458567789925530969960632022991725288820052123056769407071490581055540034033916024865616239081739199
2423103942326621055715214551153418769137416085482365363841026480692224927457609291101280463902088271667165648
3175018702509757797663212125203718817663489478696532579375620937155049862690143272747967539240842857858339607
7637703889114956927940495898241162789688812554693969512123617640884299865875135695286000835813043172034547387
5823215450528987811881575435362151225668218174595875147787505283846162146634908480514364258845351394842822471
22994950556951028152'
N[(15/16)! ,1000]
0.975165947987594223243527641276801058075345188586314791510618484313273486134580437827027051569041315313286138
1985417072506583666517654753280853186139977192467937290440722729734562708231871996559463596566585147390578461
32970091622305104625681174384377113453969487307826955129341908375486607495650864169370598118229235848416673523
25490019036652228556134851315724122803817686315162540252625515529768296381086047478542510421136748996055183139
49845743617553085091001875220956261146858158877395249554312828165511334582391238332723987906990672113654304052
76235880325111535815161455072469225560710793087334224899742732278515852548798356194799508671988039668625263475
65428512161019412476835676193441220789829408278055167756931424697362697547595297233258006523834847084130839395
89546566898149255046481266821648730990450717330549659014759909418959593340196441369069202367541573559905520029
25034882940873201658670986647950160981895736696840646314389100435243702672933133919444028794652650360771338749
2041424486092...
It was Andrew's idea to try (15/16)! Don thought that was a fine idea. Andrew worked on trying to find a repeating pattern in the factorials, but could find only repeats of a digit, but no continuing pattern.
Tuesday March 12:
1. Andrew worked on a pyramid made from wooden cm cubes. There were squares in each layer, 4 in the first layer, then 16, 36, 64, and 100 in the bottom layer. Don asked Andrew how many cubes would be in the next layer and he quickly figured out that there would be 144. Then Don asked him to find a shortcut or rule to get the number of cubes in any layer. He thought for a while and came up with this rule if L is the layer number and N is the number of cubes in that layer: (2*L)2 = N
2. Andrew spent a little time on the Chinese Ring Puzzle.
3. Don gave Andrew the problem to share 5 cookies between 3 people. He used scissors to cut the cookies. Each person would get 1 cookie, then he tried to cut the remaining 2 cookies and came up with unusual shaped pieces, ending up with each person getting 1 and 2/6 or 1 1/3 cookies. They talked about this as being impossible. He tried again and this time got 1 2/3, which was correct, because he added 1 2/3 three times and got 5, the starting number of cookies.
4. Andrew started work on another pyramid (lying around), in which the layers were in a triangular form; the following number of cubes were in each layer: 1, 3, 6, 10, 15 and 21. He realized quickly that 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, and so on. Then Don asked him to find the number of cubes in the 100th layer! He worked on that for a while, but hadn't come up with a rule before he left.