The Base That Fits

Question

A legacy market-data feed prints every integer in one fixed base, and it renders the decimal number 67 as "43". What base is the feed using? Sanity-check your answer, then decide whether some other base would render 67 as "34".

Accepted Answer

Set up the digit polynomial. A base-b numeral is a polynomial in b with the digits as coefficients. So "43" means 4b + 3 = 67 implies 4b = 64 implies b = mathbf16. Legality and sanity checks. Both digits (4 and 3) are below 16, and the leading digit is nonzero, so "43" is a valid base-16 numeral. Cross-check from the other direction: hexadecimal 43_16 = 4 cdot 16 + 3 = 67. (If you've stared at protocol dumps, you may even recognize byte 0x43 as the ASCII character "C", whose code is exactly 67.) The twist. Could "34" also spell 67? Same machinery: 3b + 4 = 67 implies 3b = 63 implies b = 21, an integer, with digits 3 and 4 both legal in base 21. So yes: the same number prints as "43" in base 16 and as "34" in base 21. Digit strings mean nothing until the base is pinned down. The general method is worth stating once: to find an unknown base, write the numeral as its digit polynomial, set it equal to the decimal value, solve, and then enforce the two legality constraints (every digit < b, leading digit ne 0). If the solve gives a non-integer or an illegal digit, no such base exists. On a desk this is more than a puzzle: exchange protocols and order IDs pack values in hex and binary, a

The Base That Fits

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