There is a way to express 64 with only two fours and no symbols beyond +, -, *, /, ^, √, !, and parenthesis, although some may be used more than once. It isn't too hard. Can you find it?

It is asserted in a reliable source that 64 can also be expressed with a single 4 using 57 square root signs, nine factorials (no double or higher factorials), and 18 sets of parentheses. I can't figure it out. Can you?

(In reply to Unreliable use of reliable source by Jer)

To abbreviate, where x is given it is a representation of the value of the number to the right of the equal sign on the line immediately above.

4! = 24

(x)! = 620448401733239439360000
SQRT(x) = 787685471322.93829008286409081066
SQRT(x) = 887516.46256446324584582484899297
SQRT(x) = 942.08092145232579582260016038814
SQRT(x) = 30.693336759829906049982869774441
SQRT(x) = 5.5401567450596473218538252721798
[x] = 5

5! = 120
SQRT(x) = 10.954451150103322269139395656016
[x] = 10

10! = 3628800
SQRT(x) = 1904.9409439665052251611633426203
[x] = 1904

1904! = 4.2472745825973511418525816898004e+5419
SQRT(x) = 6.5171117702532547112094165410681e+2709
SQRT(x) = 8.0728630424733793517929272334592e+1354
SQRT(x) = 2.8412784169231601988385890164626e+677
SQRT(x) = 5.3303643561422329836288643326462e+338
SQRT(x) = 2.30874818485657630892346873856e+169
SQRT(x) = 4.8049538862059605711263081650251e+84
SQRT(x) = 2.1920205031445213585452995514133e+42
SQRT(x) = 1480547366059094158876.7135167089
SQRT(x) = 38477881517.296325274268941349014
SQRT(x) = 196157.79749297840562320658846463
SQRT(x) = 442.8970506708962107256131677764
[x] = 442

442! = 1.0974001126960778177256496689178e+979
SQRT(x) = 3.3127029940761031312139878789565e+489
SQRT(x) = 5.7556085638932249500461753324919e+244
SQRT(x) = 2.399084943034161295856419440326e+122
SQRT(x) = 1.5488979769610912936375103655053e+61
SQRT(x) = 3935604117490847546180439715890.9
SQRT(x) = 1983835708291099.2773842391505631
SQRT(x) = 44540270.635584369755916927227169
SQRT(x) = 6673.8497612385888766533174715788
[x] = 6673

6673! = 9.1000611134476162069455859819878e+22623
SQRT(x) = 9.5394240462659045204889697072316e+11311
SQRT(x) = 9.7669975152376815378883202551354e+5655
SQRT(x) = 9.8828121075115465021570060862174e+2827
SQRT(x) = 9.9412333779624882630719713967753e+1413
SQRT(x) = 9.97057332927204448731969512968827e+706
SQRT(x) = 3.1576214771122337413377611891934e+353
SQRT(x) = 5.6192717296036091774138573476409e+176
SQRT(x) = 2.3705003120867985191402757690749e+88
SQRT(x) = 1.5396429170709676340281047660596e+44
SQRT(x) = 12408234834459604286958.937503027
SQRT(x) = 11139225618.04147276173163891005
SQRT(x) = 333754.7851612639866883425637369
SQRT(x) = 577.7151418833194964748975184999
[x] = 577

577! = 2.456228105509100896974185441324e+1344
SQRT(x) = 1.5672358168154213342494549626166e+672
SQRT(x) = 1.2518928935078357310362803668778e+336
SQRT(x) = 1.118880196226493111951094555398e+168
SQRT(x) = 1.0577713345645613173859961468333e+84
SQRT(x) = 1.0284801089785651840863830918776e+42
SQRT(x) = 1014140083508469348621.4576598044
SQRT(x) = 31845566151.482836349562886577302
SQRT(x) = 178453.26041146694708815585109304
SQRT(x) = 422.43728577324581031033608684725
[x] = 422

422! = 2.0961138301257663587685640189391e926
SQRT(x) = 1.4477961977176782041421200163017e+463
SQRT(x) = 3.8049917184110640375974960952611e+231
SQRT(x) = 6.1684614924720605226526167163405e+115
SQRT(x) = 7.8539553681390757713882405330482e+57
SQRT(x) = 88622544356044505135287872665.99
SQRT(x) = 297695388536746.57675111679091909
SQRT(x) = 17253851.4116920160971831932696
SQRT(x) = 4153.7755610639359744380644282928
SQRT(x) = 64.4497910086909123907200805208737
[x] = 64

Posted by Dej Mar on 2006-04-11 16:22:20