# Functions

## by Michael Korn

Each letter represents some function, as described below. Each line of the puzzle consists of an input/output pair, followed by a complete list of all the functions for which that relation holds. Using the given list, solvers have to determine what the rule for each letter is. Following the given equations are several equations where the values have been replaced with question marks. Solvers have to deduce the unique input/output pair which works for the indicated set of rules, and no others. Having done that, they can take the numbers in parentheses, convert them to letters in the usual way (1=A, 2=B, etc.) to produce the answer word.

Here are the rules for each letter:

 A. HALF. The output is x/2. B. PRODUCT OF DIGITS. The output is the product of the digits of x. C. DIVIDE BY LAST DIGIT. The output is x divided by the last digit of x. D. NUMBER OF LETTERS. The output is the number of letters in the English name of the number x. E. NUMBER OF FACTORS. The output is the number of factors of x. F. SUM OF DIGITS. The output is the sum of the digits of x. G. FLOOR OF SQUARE ROOT. The output is the square root of x, rounded down. H. EULER'S TOTIENT. This is Euler's totient function, a.k.a. the Euler Phi function. The output is the number of positive integers less than or equal to x which are relatively prime to x. I. EIGHT. The output is 8. J. TRIANGLE INVERSE. This function takes the nth triangular number to n. To be precise, the output is (sqrt(1+8x)-1)/2. K. BINARY DENSITY. The output is the number of 1's in the binary representation of x. L. XTH DIGIT OF PI. The output is the xth decimal digit of pi, treating the "3" before the decimal point as the first digit. M. ORDINAL OF FIRST LETTER. The output is the first letter in the English name of the number x, converted to a number in the range 1-26 in the usual way. N. FLOOR OF NATURAL LOG. The output is the natural log of x, rounded down. O. NUMBER OF DISTINCT PRIME FACTORS. The output is the number of distinct prime factors of x. P. AVERAGE OF DIGITS. The output is the mean of the digits of x. Q. LENGTH AS ROMAN NUMERAL. The output is the number of letters in the Roman numeral representation of x. R. FAHRENHEIT TO CELSIUS. This function performs the Fahrenheit to Celsius conversion on x. The output is 5/9*(x-32).

Here are the solutions to the unknown equations. In some cases there are multiple input/output pairs which work for all the listed letters, but there is only one pair which works for the given letters and no others.

```41 -> (5)     F,R
(21) -> 6     J,L
(18) -> 9     A,F
(15) -> 2     N,O,Q
6 -> (3)      A,D,J
(5) -> 4      D,H
(14) -> 3     G,K,Q
225 -> (20)   B,D,M
(18) -> 6     E,H
81 -> (9)     D,F,G,L
15 -> (3)     C,G,P
```

When converted to letters, the numbers in parentheses spell the answer, EUROCENTRIC.