Example I
No. in right hand. 18 3 No. in left hand. 7 2 - 54. 54 14 - - 14 68 sum of the products.
No in right hand No. in left hand. 7 18 3 2 57 sum of the products.
A Person having fixed on a Number in his Mind, to tell him u hai Number it is.
Bid him quadruple the number thought on, or multiply it by 4; and having done this, desire him to add 6, 8, 10, or any even number you please, to the product; then let him take the half of this sum, and tell you how much it is; from which, if you take away half the number you desired him at first to add to it, there will remain the double of the number thought on.
EXAMPLE.
Suppose the number thought on is The quadruple of it is 8 added to the product is And the half of this sum 4 taken from this leaves Therefore 5 was the number thought on. .... 5 20 ... 28 14 10.- ....
Another Method of discovering a Number thought on.
After the person has fixed on a number, bid him double it, and add 4 to that sum; then let him multiply the whole by 5, and to that product add 12; desire him also to multiply this sum by 10, and after having deducted 302 from the product, to tell you the remainder, from which, if you cut off the last two figures, the number that remains will be the one thought on.
EXAMPLE.
Let the number thought on be Then the double of this is 7 .. 14 18 ....... 90 102 ... .............. 1020 302 718,- And 4 added to it makes .....
This multiplied by 5 is .......
And 12 added to it is And this multiplied by 10 is From which deducting There remains which, by striking off the last two figures, gives 7, -the number thought on. To tell the Number a Person has fixed upon, without asking him any Questions.
The person having chosen any number in his mind, from l to 15, bid him add one to it, and triple the amount. Then, If it be an even number, let him take the half of it, and triple that half; but if it be an odd number, he must add 1 to it, and then halve it, and triple that half.
In like manner let him take the half of this number, if it be even, or the half of the next greater, if it be odd; and triple that half.
Again, bid him take the half of this last number, if even or of the next greater, if odd; and the half of that half in the same way; and by observing at what steps he is obliged to add 1 in the halving, the following table will shew the number thought on: 1-0-0 2-0-0 3-0-0 1-2-0 1-3-0 1-2-3 2-3-0 0-0-0 - 4-8 --13-5 --- 3-11 - 2-10 - 8-0 - 6-14 - 1-9 -15-7 Thus, if he be obliged to add 1 only at the first step, or halving, either 4 or 8 was the number thought on; if there were a necessity to add 1 both at the first and second steps, either 2 or 10 was the number thought on, &c.
And which of the two numbers is the true one may always be known from the last step of the operation; for if 1 must be added before the last half can be taken, the number is in the second column, or otherwise in the first, as will appear from the following examples:
Suppose the number chosen to be ........ . 9 To which, if we add ........
The sum is Then the triple of that number is 1. The half of which is ...
The triple of 15 is 2. And the half of that is ......
The triple of 23 is 3. The half of that is ... ... 1 10 30 .. 15 45 .. 23 69 35 18 And the half of that is From which it appears, that it was necessary to add 1 both at the second and third steps, or halvings; and therefore, by the table, the number thought on is either 1 or 9. And as the last number was obliged to be augmented by 1 before the half could be taken, it follows also, by the above rule, that the number must be in the second column; and consequently it is 9. Again, suppose the number thought on to be To which, if we add Then the triple of that number is The sum is 1. The half of which is ..
The triple of 11 is .... .... 6 1 7 21 11 ..... .. 33 2. And the half of that is The triple of 17 is 3. The half of that is And the half of that half is From which it appears, that it was necessary to add 1 at all the steps, or halvings, 1, 2, 3, therefore, by the table, the number thought on is either 6 or 14.
And as the last number required no augmentation before its half could be taken, it follows also, by the above rule, that the number must be in the first column; and consequently it is 6.
A curious Recreation, usually called-The Blind Abbess and her Nuns.
A blind abbess visiting her nuns, who were twenty-four in number, and equally distributed in eight cells, built at the four corners of a square, and in the middle of each side, finds an equal number in every row, containing three cells. At a second visit, she finds the same number of persons in each row as before, though the company was increased by the accession of four men. And coming a third time, she still finds the same number of persons in each row, though the four men were then gone, and had each of them carried away a nun.
Fig. 1.
Fig. 2.
Fig. 3. 333 252 414 3 3 5 5 1 1 333 252 414 Let the nuns be first placed as in fig. 1, three in each cell; then when the four men have got into the cells, there must be a man placed in each corner, and two nuns removed thence to each of the middle cells, as in fig. 2, in which case there will evidently be still nine in each row; and when the four men are gone, with the four nuns with them, each corner cell must contain four nuns, and every other cell one, as in fig. 3; it being evident, that in this case also, there will still be nine in a row, as before.
Any Number being named, to add a Figure to it, which shall make it divisible by 9.
Add the figures together in your mind which compose the number named; and the figure which must be added to this sum, in order to make it divisible by 9, is the one required. Suppose, for example, the number named was 8654; you find that the sum of its figures is 23; and that 4 being added to this sum will make it 27; which is a number exactly divisible by 9 You therefore desire the person who named the number 8654, to add 4 to it; and the result, which is 8658, will be divisible by 9, as was required. -This recreation may be diversified, by your specifying, before the sum is named, the particular place where the figure shall be inserted, to make the number divisible by 9; for it is exactly the same thing, whether the figure be put at the end of the number, or between any two of its digits. 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 snewing 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 pronounce, 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.