पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/३४७

CHAPTER VI-MIXED PROBLEMS. HAPTER 149 gold of) 12 rarnas with the aid of the two pellets." O you, who know the secret of calculation, if you possess cleverness in relation to calculations bearing upon gold, tell the quickly, after thinking out well, the measures of the quantities of gold possessed by both of them, and also of the varnas (of those quantities of gold) Thus enils Suvarna-kuttikära in the chapter on mixed problems Ticitra-kuttikāra. Hereafter we shall expound the Victra-kuttikära in the chapter on mixed hoblen s The rule in regard to (the ascertaiung of) the number of truthful and untruthful statements (in a situation like the one given below wherein both are simultaneously possible):- 216. The number of men, multiplied by the number of those liked (among them) as increased by one, and (then) diminished by twice the number of men liked, gives rise to the number of untruthful statements. The square of the number representing all the men, diminished by the number of these (autruthful statements), gives rise to the statements that are truthful. 216. The rationale of this inle will be clear from the following algebraical representation of the problem given in stanza 217 below -- Let a he the total number of persons of whom baie hiked. The number of utterances is u, and each statement refers to a persons Hence the total number of st tements is a xa o1 ² Now, of these a persons, bare liked, and a-b are not liked When each of the b number of persons is told-"You alone are liked," the number of Therefore, the total number I. uatruthful statements in cach case is bl of untruthful statements in b statements is b (b-1) When, again, the same state.uent is made to each of the ab persons, the number of untinthinl statements in each case is b + 1 Therefore, the total number of untruthful statements in a--b utterances is (a - b)(b + 1) . II. Adding I and II, we got b (b-1) + (a-b) (b + 1) = a (b + 1) - 26. This represents the total of untruthful statements, and on subtracting it from a, which is the measure of all the statements, truthful and unta uthful, we arrive obviously at the measure of the truthful statements. .