Very straightforward, this one. And I don't need to feel bad that I did it the simple way. If you're wondering where split-sequence-if and read-file came from, the answer is rutils, which is a good general-purpose utility package, a sort of superset of alexandria, and just a (ql:quickload 'rutils) away. As before, the final answer is obtained by evaluating (euler42).
Statutory Warning: spoilers ahead
;; If x = 0.5 * n * (n+1) is a triangle number, then there is an
;; integral solution for n^2 + n - 2*x = 0.
(defun quadratic-solution (a b c)
(let ((det (sqrt (- (* b b) (* 4 a c))))
(denom (* 2 a)))
(cons (/ (- (- b) det) denom)
(/ (+ (- b) det) denom))))
(defun positive-int (num)
(and (integerp num)
(plusp num)))
(defun is-triangle-number (num)
(let ((soln (quadratic-solution 1 1 (* (- 2) num))))
(or (positive-int (car soln))
(positive-int (cdr soln)))))
(defun char->number (char)
(1+ (- (char-code char) (char-code #\A))))
(defun is-triangle-word (word)
(let ((value (loop for char across word
summing (char->number char))))
(is-triangle-number value)))
(defun euler42 ()
;; Read in the file, split on the commas, remove the quotes
(let* ((file-text (read-file "/home/agam/Downloads/p042_words.txt"))
(words-with-quotes (split-sequence-if
(lambda (char) (eq char #\,))
file-text))
(words (mapcar (lambda (w) (remove #\" w))
words-with-quotes)))
(loop for w in words count
(is-triangle-word w))))