_A Person having made choice of several Numbers, to tell him what Number will exactly divide the Sum of those which he has chosen._

Provide a small bag, divided into two parts; into one of which put several tickets, numbered 6, 9, 15, 36, 63, 120, 213, 309, or any others you please, that are divisible by 3, and in the other part put as many different tickets marked with the number 3 only.

Draw a handful of tickets from the first part, and, after shewing them to the company, put them into the bag again; and having opened it a second time, desire any one to take out as many tickets as he thinks proper.

When he has done this, open privately the other part of the bag, and tell him to take out of it one ticket only.

You may then p.r.o.nounce, that this ticket shall contain the number by which the amount of the other numbers is divisible; for, as each of these numbers is some multiple of 3, their sum must evidently be divisible by that number.

This recreation may also be diversified, by marking the tickets in one part of the bag with any numbers which are divisible by 9, and those in the other part of the bag with the number 9 only; the properties of both 9 and 3 being the same; or if the numbers in one part of the bag be divisible by 9, the other part of the bag may contain tickets marked both with 9 and 3, as every number divisible by 9 is also divisible by 3.

_To find the Difference between any two Numbers, the greater of which is unknown._

Take as many 9's as there are figures in the less number, and subtract the one from the other.

Let another person add that difference to the larger number; and then, if he take away the first figure of the amount, and add it to the remaining figures, the sum will be the difference of the two numbers, as was required.

Suppose, for example, that Matthew, who is 22 years of age, tells Henry, who is older, that he can discover the difference of their ages.

He privately deducts 22, his own age, from 99, and the difference, which is 77, he tells Henry to add to his age, and to take away the first figure from the amount.

Then if this figure, so taken away, be added to the remaining ones, the sum will be the difference of their ages; as, for instance:

The difference between Matthew's age and 99, is 77 To which Henry adding his age 35 ---- The sum will be 112 And 1, taken from 112, gives 12 Which being increased by 1 -- Gives the difference of the two ages 13 And, this added to Matthew's age 22 -- Gives the age of Henry, which is 35

_A Person striking a Figure out of the Sum of two given Numbers, to tell him what that Figure was._

Such numbers must be offered as are divisible by 9; such, for instance, as 36, 63, 81, 117, 126, 162, 207, 216, 252, 261, 306, 315, 360, and 432.

Then let a person choose any two of these numbers, and after adding them together in his mind, strike out any one of the figures he pleases, from the sum.

After he has done this, desire him to tell you the sum of the remaining figures; and that number which you are obliged to add to this amount, in order to make it 9, or 18, is the one he struck out.

For example, suppose he chose the numbers 126 and 252, the sum of which is 378.

Then, if he strike out 7 from this amount, the remaining figures, 3 and 8, will make 11; to which 7 must be added to make 18.

If he strike out the 3, the sum of the remaining figures, 7 and 8, will be 15; to which 3 must be added, to make 18; and so in like manner, for the 8.

_By knowing the last Figure of the Product of two Numbers, to tell the other Figures._

If the number 73 be multiplied by each of the numbers in the following arithmetical progression, 3, 6, 9, 12, 15, 18, 21. 24, 27, the products will terminate with the nine digits, in this order, 9, 8, 7, 6, 5, 4, 3, 2, 1; the numbers themselves being as follows, 219, 438, 657, 876, 1095, 1314, 1533, 1752, and 1971.

Let therefore a little bag be provided, consisting of two part.i.tions, into one of which put several tickets, marked with the number 73; and into the other part, as many tickets numbered 3, 6, 9, 12, 15, 18, 21, 24, and 27.

Then open that part of the bag which contains the number 73, and desire a person to take out one ticket only; after which, dexterously change the opening, and desire another person to take a ticket from the other part.

Let them now multiply their two numbers together, and tell you the last figure of the product, and you will readily determine, from the foregoing series, what the remaining figures must be.

Suppose, for example, the numbers taken out of the bag were 73, and 12; then, as the product of these two numbers, which is 876, has 6 for its last figure, you will readily know that it is the fourth in the series, and that the remaining figures are 87.

_A curious Recreation with a Hundred Numbers, usually called the Magical Century._

If the number 11 be multiplied by any one of the nine digits, the two figures of the product will always be alike, as appears from the following example:--

11 11 11 11 11 11 11 11 11 1 2 3 4 5 6 7 8 9 -- -- -- -- -- -- -- -- -- 11 22 33 44 55 66 77 88 99

Now, if another person and yourself have fifty counters apiece, and agree never to stake more than ten at a time, you may tell him, that if he will permit you to stake first, you will always undertake to make the even century before him.

In order to this you must first stake one, and remembering the order of the above series, constantly add to what he stakes as many as will make one more than the numbers 11, 22, 33, &c. of which it is composed, till you come to 89; after which, the other party cannot possibly make the even century himself, or prevent you from making it.

If the person who is your opponent have no knowledge of numbers, you may stake any other number first, under 10, provided you afterwards take care to secure one of the last terms, 56, 67, 78, &c.: or you may even let him stake first, provided you take care afterwards to secure one of these numbers.

This recreation may be performed with other numbers; but, in order to succeed, you must divide the number to be attained, by a number which is an unit greater than what you can stake each time; and the remainder will then be the number you first stake. Suppose, for example, the number to be attained is 52, and that you are never to add more than six; then dividing 52 by 7, the remainder, which is 3, will be the number you must stake first; and whatever the other stakes, you must add as much to it as will make it equal to 7, the number by which you divided; and so on.

_A Person in Company having privately put a Ring on one of his fingers, to Name the Person, the Hand, the Finger, and even the Joint on which it is placed._

Desire a third person to double the number of the order in which the wearer of the ring stands, and add 5 to that number, then multiply that sum by 5, and to the product add 10. Let him then add 1 to the last number, if the ring be on the right hand, and 2 if on the left, and multiply the whole by 10: to this product he must add the number of the finger, beginning with the thumb, and multiply the whole again by 10.

Desire him then to add the number of the joint; and lastly, to increase the whole by 35.

This being done, he is to declare the amount of the whole, from which you are to subtract 3535; and the remainder will consist of four figures, the first of which will give the place in which the person stands, the second the hand, 1 denoting the right, and 2 the left hand, the third number the finger, and the fourth the joint.

EXAMPLE.

Suppose the person stands the second in order, and has put the ring on the second joint of the little finger of the left hand:

Double the order is 4 Add 5 -- 9 Multiply by 5 -- 45 Add 10 -- 55 Number for left hand 2 -- 57 Multiply by 10 ---- 570 Number of finger 5 ---- 575 Multiply by 10 ---- 5750 Number of joint 2 ---- 5752 Add 35 ---- 5787 Subtract 3535 ---- 2252

Hence it will appear that the first 2 denotes the second person in order, the second 2 the left hand, 5 the little finger, and 2 the second joint.

_To make a Deaf Man hear the Sound of a Musical Instrument._

It must be a stringed instrument, with a neck of some length, as a lute, a guitar, or the like; and before you begin to play, you must by signs direct the deaf man to take hold with his teeth of the end of the neck of the instrument; for then, if one strikes the strings with the bow one after another, the sound will enter the deaf man's mouth, and be conveyed to the organ of hearing through a hole in the palate, and thus the deaf man will hear with a great deal of pleasure the sound of the instrument, as has been several times experienced; nay, those who are not deaf may make the experiment upon themselves, by stopping their ears so as not to hear the instrument, and then holding the end of the instrument in their teeth, while another touches the strings.

_When two Vessels or Chests are like one another, and of equal Weight, being filled with different Metals, to distinguish the one from the other._

This is easily resolved, if we consider that two pieces of different metals, of equal weight in air, do not weigh equally in water, because that of the greatest specific gravity takes up a lesser s.p.a.ce in water; it being a certain truth, that any metal weighs less in water than in air, by reason of the water, the room of which it fills; for example, if the water weighs a pound, the metal will weigh in that water a pound less than in the air: this gravitation diminishes more or less, according as the specific gravity of the metal is greater than that of the water.

We will suppose, then, two chests perfectly like one another, of equal weight in the air, one of which is full of gold, and the other of silver; we weigh them in water, and that which then weighs down the other must needs be the gold chest, the specific gravity of gold being greater than that of silver, which makes the gold lose less of its gravitation in water than silver. We know by experience, that gold loses in water about an eighteenth part only, whereas silver loses near a tenth part; so that if each of the two chests weighs in the air, for example, 180 pounds, the chest that is full of gold will lose in the water ten pounds of its weight; and the chest that is full of silver will lose eighteen: that is, the chest full of gold will weigh 170 pounds, and that of silver only 162.