therefore of throwing them, at one throw, is expressed by and, as the adventurer was allowed 20 throws, the probability of his succeeding was only which is nearly equal to To play an equal game therefore, the mountebank should have engaged to return 2332 times the money deposited.

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PROBLEM VIII.

The same mountebank offered a new chance to the person who had lost, on the following conditions: to deposit a sum equal to the former, and to receive both the stakes in case he should bring all the blank faces, in 3 successive throws.

Those unacquainted with the method to be pursued in order to resolve such problems, are liable to reason in an erroneous manner respecting dice of this kind; for, observing that there are five times as many blank as marked faces, they thence conclude that it is 5 to 1 that the person who throws them will not bring any point. They are, however, mistaken, as the probability, on the contrary, is 2 to 1 that they will not come up all blank.

If we take only one die, it is 5 to 1 that the person who holds it will throw a blank; but if we add a second die, it may be readily seen, that the marked face of the first may combine with each of the blank faces of the second, and the marked face of the second with each of the blank faces of the first; and, in the last place, the marked face of the one with the marked face of the other: consequently, of the 36 combinations of the faces of these two dice, there are 11, in which there is at least one marked face. But, as we have already observed, this number 11 is the difference of the square of 6, the number of the faces of one die, and of the square of the same number diminished by unity, that is to say of 5.

If a third die be added, we shall find, by the like analysis, that, of the 216 combinations of three dice, there are 91 in which there is at least one marked

face; and 91 is the difference of the cube of 6 or 216, and the cube of 5 or 125; the result will be the same in regard to the more complex cases; and hence we may conclude, that of the 46656 combinations of the faces of the 6 dice in question, there will be 31031 in which there is at least one marked face, and 15625 in which all the faces are blank; consequently, the chance is 2 to 1 that some point, at least, will be thrown; whereas, by the above reasoning, it would appear that 5 to 1 might be betted on the contrary being the

case.

PROBLEM IX.

In how many throws, with six dice, marked on all their faces, may a person engage, for an even bet, to throw 1, 2, 3, 4, 5, 6?

We have just seen that there are 46655 chances to 1 that a person will not throw these 6 points with dice marked only on one of their faces; but the case is very different with 6 dice marked on all their faces; and to prove it, we need only to observe, that the point 1, for example, may be thrown by each of the dice, as well as the 2, 3, &c. which renders the probability of these six points, 1, 1, 3, &c. coming up, much greater.

But to analyze the problem more accurately, we shall observe, that there are 2 ways of throwing 1, 2, with 2 dice; viz. 1 with the die A, and 2 with the die B; or 1 with the die B, and 2 with A. If it were proposed to throw 1, 2, 3, with 3 dice; of the whole of the combinations of the faces of 3 dice, there are 6 which give the points 1, 2, 3; for 1 may be thrown with the die A, 2 with B, and 3 with C; or 1 with A, 2 with C, and 3 with B; or 1 with B, 2 with A, and 3 with C; or 1 with B, 2 with C, and 3 with A; or 1 with C, 2 with A, and 3 with B; or 1 with C, 2 with B, and 3 with A.

It hence appears, that to find the number of ways in which 1, 2, 3, can be thrown with 3 dice, 1, 2, 3 must be multiplied together. In like manner, to find the

number of ways in which 1, 2, 3, 4 can be thrown with 4 dice, we must multiply together 1, 2, 3, 4, which will give 24; and, in the last place, to find in how many ways 1, 2, 3, 4, 5, 6 can be thrown with 6 dice, we must multiply together these six numbers, the product of which will be 720.

If the number 46656, which is the combinations of the faces of 6 dice, be divided by 720, we shall have 644 for the chances to 1, that these points will not come up at one throw; and, consequently, a person may undertake for an even bet to bring them in 64 throws.

In the last place, as the dice may be thrown 130 times, and more, in a quarter of an hour, a person may, with advantage, bet more than 2 to 1, that they will come up in the course of that time.

He who engages for an even bet to throw these points, in a quarter of an hour, undertakes what is highly advantageous to himself, and equally disadvantageous to his adversary.

ARITHMETICAL AMUSEMENTS IN DIVINATION AND COMBINATION.

PROBLEM I.

To tell the number thought of by a person.

Desire the person, who has thought of a number, to triple it, and to take the exact half of that triple if it be even, or the greater half if it be odd. Then desire him to triple that half, and ask him how many times it contains 9; for the number thought, if even, will contain twice as many units as it does nines, and one more if it be odd.

Thus, if 5 has been the number thought of, its triple will be 15, which cannot be divided by 2 without a remainder. The greater half of 15 is 8; and if this half be multiplied by 3, we shall have 24, which contains 9 twice; the number thought of will therefore be 4 plus 1, that is to say 5.

II. Bid the person multiply the number thought of by itself; then desire him to add unity to the number thought of, and to multiply it also by itself; in the last place, ask him to tell the difference of these two products, which will certainly be an odd number, and the least half of it will be the number required.

Let the number thought of, for example, be 10, which multiplied by itself gives 100; in the next place, 10 increased by 1 is 11, which multiplied by itself makes 121; and the difference of these two squares is 21, the least half of which being 10, is the number thought of.

This operation might be varied by desiring the person to multiply the second number by itself, after it has been diminished by unity. In this case, the number thought of will be equal to the greater half of the difference of the two squares.

Thus, in the preceding example, the square of the number thought of is 100, and that of the same number less unity is 81: the difference of these is 19, the greater half of which, or 10, is the number thought of.

III. Bid the person take 1 from the number thought of, and then double the remainder; desire him to take 1 from this double, and to add to it the number thought of: in the last place, ask him the number arising from this addition, and if you add 3 to it, the third of the sum will be the number thought of.

The application of this rule is so easy that it is needless to illustrate it by an example.

IV. Desire the person to add 1 to the triple of the number thought of, and to multiply the sum by 3; then bid him add to this product the number thought of, and the result will be a sum, from which if 3 be subtracted, the remainder will be decuple of the number required. If 3 therefore be taken from the last sum, and if the cipher on the right be cut off from the remainder, the other figure will indicate the number sought.

Let the number thought of be 6, the triple of which is 18; and if unity be added it makes 19; the triple

E

of this last number is 57, and if 6 be added it makes 63, from which if 3 be subtracted the remainder will be 60 now if the cipher on the right be cut off, the remaining figure 6 will be the number required.

V. Another method of telling the number any one has thought of.

These operations, by which a person seems to guess the thoughts of another, may be introduced very opportunely in company, when any one asserts that all amusing tricks are performed by slight of hand. The following method may be found in Ozanam, but we have here made some additions to it. 1st. Desire any person to think of a number, but that we may not speak in too abstract a manner, it will be best to desire him to think of a certain number of guineas. 2d. Tell the person that some one of the company lends him a similar sum, and request him to add them together, that the amount may be known. It will here be

proper to name the person who lends him a number of guineas equal to the number thought of, and to beg the one who makes the calculation to do it with great care, as he may readily fall into an error, especially the first time. 3rd. Then say to the person, I do not lend you, but give you 10, add them to the former sum. 4th. Continue in this manner:-Give the half to the poor, and retain in your memory the other half. 5th. Then add:-Return to the gentleman, or lady, what you borrowed, and remember that the sum lent you was exactly equal to the number you thought of. 6th. Ask the person if he knows exactly what remains; he will answer Yes: you must then say, And I know also the number that remains, it is equal to what I am going to conceal in my hand. 7th. Put into one of your hands 5 pieces of money, and desire the person to tell how many you have got. He will reply 5: upon which open your hand, and shew him the 5 pieces. You may then say I well knew that your result was 5; but if you had thought of a very large number, for example, two or three millions, the result would have been much