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Hidden Gems: Crazy Calculations
Learn how to conjure with a calculator courtesy of eighteenth-century magician Professor Pinetti!
If you have been keeping up with the articles published in the Ruseletter lately, you may be aware that I have been researching the flamboyant Italian Conjurer Professor Pinetti. In his lifetime, he wrote and published only one book, which included three tricks that involved mathematical calculations. Personally, I like puzzling mental magic tricks using numbers and, as they only call for a pen and paper, they make fantastic impromptu performance pieces, especially for individuals working in an office. However, these tricks can sometimes put a spectator under pressure to add, subtract, or multiply numbers in public. Although the calculations involved are usually simple, many people feel anxious about mathematics and find it distressing to do it, especially in front of other people.
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In today's world, smartphones have become ubiquitous.1 Although I am not particularly enthusiastic about their excessive use, universal presence and pervasive nature, there is a positive aspect to their omnipresence. The majority of individuals now carry a pocket calculator with them at all times, thanks to the built-in calculator apps found on most smartphones. This feature has made computation-related tasks more manageable and eliminates the need to rely on spectators to perform mental arithmetic, making these magician-spectator interactions less awkward.
In this edition of Hidden Gems, I’ve extracted three of these calculation tricks from Giovanni Giuseppe Pinetti’s 1784 book Physical Amusements and Diverting Experiments.
A Puzzling Question
In Chapter IX, there’s a trick called “A puzzling Question to be proposed for Solution.” Here’s the text from the book:
Set down three sums on paper; and say to the company, ladies and gentlemen, there are three sums, very different from each other, and very disproportionate; yet I wish to divide them among three persons, so that they may have an equal sum each, and yet without altering any thing in either of the sums. This will appear very difficult, yet nothing so simple and easy; one single addition will suffice to prove to you that the amount of each sum will be the same, and that the shares will not enrich much the respective persons: here is the proof:
Cast up the first of these sums in the following manner, and say, 5 and 1 make 6; 3 more, 9; 4 more, 13; 1 more, 14; 2 more, 16; and 2 more, 18; set down - 18
Make the addition of the second sum in the same manner as you have done the first, and you will find the same sum of - 18
Then proceed for the third as in the two preceding, and the product will be also - 18
Here then is my division made, and each person will have only 18, as I have proved by the foregoing example.
By this we see, that nothing more is required than to be attentive in setting the sums, to make the numbers so that each sum may amount only to 18.
You may make the same question on whatever sum you please, only observing, as above, that the amount of the numbers you set may not exceed the sum you desire to belong to each person that is to have a share.
This is the weakest of the three calculation tricks in the book, and I don’t recommend that you perform it (at least not as described by Pinetti). It may function as a bar bet, but as a magic trick, it doesn't really work. However, the principle could be used differently to facilitate the force of a two-digit number, such as 18. Write several number strings on small slips of paper, making sure that each string reduces to 18 when added together. Show your audience that each slip of paper has a different number written on it, then place them in a small bag or box. Have someone freely select a slip and then "remove" one at random. In reality, you have a slip palmed in your hand that adds up to a different random number, such as 27. When you remove the slip from the bag, you actually remove the palmed slip instead. Demonstrate what you want your participant to do by adding up the numbers on your slip and announcing the sum total.
It is assumed that each number written on the paper slips corresponds to a unique two-digit number. However, one alternative approach to make the ruse more convincing is to incorporate a switching mechanism, like a concealed compartment within a box or bag. This will allow for the introduction of a second batch of number slips that truly produce distinct two-digit numbers. Checking other slips will then produce different results, making it more challenging for someone to deconstruct the method. The big benefit here is that all the slips look different but still force the same two-digit number.
This idea could be used in many routines. You could use it to read someone’s mind or force another object, such as a playing card eighteenth from the top of the pack or the eighteenth item on a written list.
A Marvelous Mathematical Prediction
Like many tricks involving some kind of calculation, this one relies on the “mathemagical” properties of the number nine. In Chapter XIX, you’ll find a trick with the title “To make an addition before the Figures are set, by knowing only how many Figures are in each Row; as likewise how many Rows compose the whole; and then adding yourself some Figures equal to those that had been set.” Despite the dry description, this is a very good prediction effect. Here’s Pinetti’s explanation of it:
Suppose the person had set five rows of figures, each row containing five figures.
Say in your mind, as you are making the addition beforehand, 9 times 5 make 45; set down 5 and carry 4: repeat the same thing for each of the five figure, as if they all counted 9; therefore for the second, say again, 9 times 5 make 45, and 4 carried over make 49; set down 9 and carry 4: in the same manner for the third, say 9 times 5 are 45, and 4 carried over are 49; set down 9 and carry 4: for the fourth do the same; and set down 9 and carry 4: for the fifth repeat the same, by setting down 9 and carrying 4.
Thus your addition being made beforehand will produce the sum of 499995: then show this addition to every body in the company; and beg some one to do you the favour of laying on a paper 5 rows of numbers, containing five figures in each row.
Suppose the numbers set for you are the following:
You ask leave to add a like quantity of numbers; in doing this, you take care that each of the figures you set down make 9 with each of the figures that have been given for you.
Total = 499995
The first figure being 2, you must set 7; the second being 9, (which completes the number wanted) you must set a cypher (0); the third being the same, operate as before; the fourth being 7, set down 2; the fifth being 1, set down 8.
The second row beginning by 1, your first figure will be 8; the second number being 4, set down 5; the third being 5, put down 4; the fourth being 6, you must set down 3; the fifth being 3, set down 6.
As the third row begins by 7, begin yours by 2; under the 6 lay 3, then 1 under the 8, and 7 under the 2.
For the fourth row, set 6 under the 3, 2 under the first 7, and another 2 under the other 7; a 0 under the 9, and 2 under the 7, which complete this row.
You are to do the same for the fifth row, putting 1 under the 8, 9 under the 0, 8 under the 1, 6 under the 3, and 9 under the 0.
Then desire some of the company to cast up these ten sums, and it will be found that the product of the whole addition will form the sum of 499995.
In order to come to this combination, you need only fix the number of figures that will compose each row, and determine the number of rows; then to reckon each row for 9, as has been shown above.
You may likewise present this addition, by saying, that it is the total amount of ten rows, composed of five figures each; out of which five rows will be set by the person who chooses to do it; then multiply secretly as many times 9 as you are to set rows of five figures; therefore multiply 5 times 9 by 5, which will give you the sum of 499995.
The person having set his numbers, you are to add your five rows, taking care that every number you set will make 9 with that to which it corresponds; which being done, you are to ask any one to cast the whole sum up, and the product will be the same as the sum you set down beforehand.
This can be presented as a simple prediction, or you can reduce it to a two-digit number to force the forty-fifth card from the top of the pack (4+9+9+9+9+5 = 45). The number of rows chosen changes the final number generated, as follows:
99999 - Two Rows
9999 - Three Rows
9999 - Four Rows
9999 - Five Rows
9999 - Six Rows
9999 - Seven Rows
9999 - Eight Rows
9999 - Nine Rows
Looking at the above list, you can observe a distinct numerical pattern. When disregarding a case with two rows, each number begins with a digit that is one less than the number of rows, followed by another digit that, when combined with the first, equates to nine. Instead of mentally computing the predicted number, as Pinetti suggests in his description, it is simpler to recall this correlation and apply it to derive the correct number once your participant has selected a number of rows. For example, if your spectator picks seven rows, you minus one from seven to get six. Then add four nines to make 69999. Finally, add a three to the end (9 - 6) to make 699993. Furthermore, if you combine all the digits in each string and form a two-digit number, the resulting figures all add up to 45. This might be useful if you want to force the forty-fifth card from the top of a pack of cards.
Using a different number of digits also changes the final total in a predictable fashion. For example, assuming three rows of numbers, here’s what the digit length does to the pattern:
297 (18) - Two Digits
2997 (27) - Three Digits
29997 (36) - Four Digits
299997 (45) - Five Digits
2999997 (54) - Six Digits
29999997 (63) - Seven Digits
299999997 (72) - Eight Digits
2999999997 (81) - Nine Digits
The trick’s biggest flaw, however, is the inclusion of four nines in the prediction number, which might seem suspicious to those who are mathematically minded. However, a newer method of performing this trick has been developed to address this concern.
To begin, ask someone to write a five-digit number on a piece of paper. Let's say they write 49358. Next, you make a prediction by adding the number two to the beginning of their number to create a six-digit number (249358). Then, subtract two from the last digit of this six-digit number. In this example, 8 - 2 = 6. Therefore, your final prediction should be 249356. This technique is superior to Pinetti's approach because there is no discernible pattern in the resulting number.
To proceed, kindly request your participant to jot down an additional five-digit number under the initial one, for instance, 57663. Afterwards, you will generate another seemingly arbitrary number. However, it is not random as it appears because each digit, when added to the one immediately above it, equals nine. Allow me to elaborate further: If the first digit of the preceding number is five, then your first digit ought to be four since 9 - 5 = 4. Similarly, the second digit should be two as it combines with the above digit to make nine (2 + 7 = 9). I hope you understand the concept; it is the same principle used by Pinetti in his trick. Based on this, your first "random" number should be 42336.
Request your assistant to provide a third five-digit number, such as 34794. Lastly, include another "random" number using the identical secret system as previously utilised. Your second number should be 65205. This will result in the completed sum as follows:
49358 - Spectator’s First Number
57663 - Spectator’s Second Number
42336 - Your First Number
34794 - Spectator’s Third Number
65205 - Your Second Number
Ask your volunteer to use the calculator app on their smartphone to add up the five numbers and read out the total. It will match your prediction exactly! In case you harbour any doubts, please attempt to add up the aforementioned five-number sum. You will find that it equals 249356.
This same prediction technique can be applied to numbers of different lengths. It functions for numbers with a greater amount of digits, such as six, seven, eight, or nine, as well as those with fewer digits, like two, three or four. Nonetheless, a five-digit number is ideal as it strikes a good balance between having too many digits, which could slow down the final calculation, and having too few, which could diminish the wow factor of the trick (larger numbers make the process feel more impossible).
I highly recommend taking some time to explore the intriguing principle behind this simple yet fantastic trick.
In Chapter XXXI, Pinetti shares another calculation-based trick called “To guess, by smelling, which has been the Number struck out by a Person in the Company, in the Product of a Multiplication given him to do.” Quite why the magician must smell the piece of paper the sum is written on is beyond me! But the trick itself uses another interesting mathematical principle. Here’s the extract from the book:
Propose to a person of the company to multiply, by whatever number he pleases, one of the three sums which you will give him on a piece of paper; desire him to strike out whatever figure he pleases of the product of his multiplication, let him change and invert the norder of the remaining figures after the defalcation he has chosen.
While the person is making his calculation and the subsequent operations, go in another room: when you are told you may return, desire the person who has done the multiplication, to give you the remaining product on a piece of paper or card; put it to your nose as though you would smell it; then you will tell him, to the great astonishment of the whole Company, what figure he had struck out.
In order to do this operation, first observe, that the figures composing each of the three sums you propose to be multiplied, do not exceed the number of 18.
Suppose the three sums proposed to be the following:
Supposing that the sum chosen to be multiplied be that of 132354. And that the multiplicator be 7. The product will then be 926478.
Suppose likewise that the figure which has been struck out is the 6, the remaining ones will form a sum of 92,478.
As you let the person who has done the multiplication set down the figures in the order he pleases, suppose also that he sets them down thus, on the piece of paper he gives you:
When you pretend to smell the paper, add together in your mind the figures presented to you, in order to reduce them to nines; and say in your mind 7 and 2 make nine; after that 8 and 4 make 12; in 12 there is 9, and three remains towards 9 more; to complete which 6 is wanting, which is and must be the figure struck out. This calculation must be made quickly, and while you pass the paper under your nose under the pretest of smelling it.
There is another manner of proceeding to guess the figure left out, by letting the person choose the sum he pleases to be multiplied, but then you must ask him to show you the sum he means to have multiplied, and to let you add one figure at your option.
In that case, by running your eyes over the sum set down, you will easily see what figure you are obliged to add in order to complete the number of 9.
In the supposition that the sum set down is the following:
Add in your mind thus: 7 and 8 are 15, and 9, 24; and 7, 31; and 8, 39; and 8 more, 47: in 47 there is 5 btimes 9, as 9 times 5 make 45; there remains 2, therefore in order to complete 9, 7 are to be added; consequently the sum to be multipled will be 7,897,887.
Then give this sum, which has been encreased by a 7, to the person who has presented it to you: and tell him to choose whatever multiplier he pleases; then retire while he does the multiplication, recommending him to strike out the figure he pleases, as usual, and to set down on a piece of paper the remaining sum, the figure being defalcated, and the remaining figures ranged as he pleases; and in order to guess the number that was struck out, you are to proceed as it has been explained for the first manner of operating, and with the same tricks.
This particular trick also utilises an intriguing principle and provides an impressive effect: the ability to discern the missing number from a given sum. The more times this trick is repeated in front of the same audience, the more perplexing it becomes. Additionally, you can incorporate this trick with the first one mentioned by utilising one of the "random" numbers written on a slip of paper to force the number 18. You can then proceed to read your audience’s mind to complete the trick.
Give Them a Go!
I hope you give these three historical magic tricks a go. Even though they were published more than two hundred years ago, they’re still surprisingly effective!
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I've made a conscious decision not to own a smartphone and have no plans of changing my mind. There are several reasons why I've arrived at this choice (too many to mention here). Not having a smartphone helps me avoid falling victim to the “attention economy”. The tech giants based in Silicon Valley are always competing for our eyeballs, often at the expense of our well-being and mental health. This is one of the reasons I prefer to publish my content on Substack, a platform that refrains from utilising manipulative algorithms that prey on our lizard brains.