पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३४५

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CHAPTER VI--MIXED PROBLEMS.

gold of) 12 varņas 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 me quickly, after thinking out well, the measures of the quantities of gold possessed by both of them, and also of the varņas (of those quantities of gold).

Thus ends Suvarna-kuțțīkāra in the chapter on mixed problems

 

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Vicitra-kuțțīkāra

Hereafter we shall expound the Vicitra-kuțțīkāra in the chapter on mixed problems.

The rule in regard to (the ascertaining 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 those (untruthful statements), gives rise to the statements that are truthful.

 

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216. The rationale of this rule will be clear from the following algebraical representation of the problem given in stanza 2]7 below:-

Let a be the total number of persons of whomb are liked. The number of utterances is a, and each statement refers to a persons. Hence the total number of statements is $a\times a{\text{ or }}a^{2}$.

Now, of these a persons, b are liked, and $a-b$ are not liked. When each of the b number of persons is told — "You alone are liked," the number of untruthful statements in each case is $b-1$. Therefore, the total number of untruthful statements in b statements is $b(b-1)$ . . . . . . . . . . I

When, again, the same statement is made to each of the $a-b$ persons, the number of untruthful 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 get $b(b-1)+(a-b)(b+1)=a(b+1)-2b$.

This represents the total of untruthful statements; and on subtracting it from $a^{2}$, which is the measure of all the statements, truthful and untruthful, we arrive obviously at the measure of the truthful statements.