Math Chat: On Four 4s And Speed Limits

Old 'four 4s' challenge (Howard Sheldon)

Can you get past 30 in forming numbers 0, 1, 2, 3, ... from four 4s and the standard mathematical operations +, - , x, /, decimal point, square root, powers, and factorial(!)?

Answer

Eric Brahinsky, Dick Feren, and Paul Goodrich got 31 = 4! + (4!+4)/4 and 32 = (4+4) x 4. Roger Bliss, Robert Lewis, Mike Soskis, and Richard Thorne also got 33 = 4!+(4- .4)/.4 and made it to 36. William Foster, William Hasek, and Michael Stern got 37 = 4! + (4! + 4)/4 and made it to 72. John Gordon got 73 by bending the rules, using .4 with a bar over it to denote .444... = 4/9 and noticing that 73 = (4!+4!+(4/9))/(4/9). I do not know whether it is possible to get 73 without bending the rules. Gordon continues on to 112.

Foster asks what is the largest prime number you can get with four 4s. His best so far is 257 = 44 + 4/4.

More marching ants

The April 25 column explained that ants marching in a single column, then two by two, and so on until 10 by 10, with one ant dropping out at each change, must start as one less than a multiple of 2520. Robert Schmidt interprets "two by two" as blocks of 4, "three by three" as blocks of 9, "four by four" as blocks of 16, and so on, and writes, "I am not sure that the problem has a solution. If it does, I'm quite sure that the answer is a very large number. I would really like to know."

Answer

Schmidt is right, this problem has no solution. If you start with n ants, then n-1 must be a multiple of 4, while n-3 must be a multiple of 16 and hence a multiple of 4. But n-1 and n-3 cannot both be multiples of 4.

New traffic capacity challenge (Marc Abel)

When a state increases a highway speed limit from 55 to 65 miles per hour, by what percent does the road capacity (in cars per hour) change?

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