At the end of a recent episode of "Penn and Teller Fool Us" Penn and Teller performed a magic trick revolving around an interesting number pattern. Penn displayed a board with the digits 1, 2, 3, 4, 5 & 6 written out in a mysterious spiral pattern. He had a guest write down a three digit number between 123 and 456 using three unique numbers from the given digits one through six. He pointed out there are 120 (6 x 5 x 4 = 120) possible numbers to choose from alluding to no way of knowing which was chosen. He instructed her to choose a second three digit number with the three remaining digits not included in the first number. There are now two numbers, each with three unique digits using digits 1, 2, 3, 4, 5 & 6 exactly once between the two of them.
Penn asked the guest to add the two numbers together to get a larger three or four digit number that may or may not have repeating digits. Lastly he asked her to add the individual digits of that larger number together (find the digit sum) and repeat the process with the new number until the answer only had a single digit (ex: 856 = 8 + 5 + 6 = 19 = 1 + 9 = 10 = 1 + 0 = 1). Penn explained it had to be a number between 1 and 9 then asked the guest to find the corresponding letter of the alphabet (1 = a, 2 = b, 3 = c, etc.). The guest then looked up the first word in the dictionary starting with that letter of the alphabet. Penn went back to his board and started to open a letter with his prediction when Teller drove through the board and onto the stage in a Taxi Cab confirming the guest's word from the dictionary.
Magic tricks are often called illusions because everything happening is real but there are hidden tricks, props and patterns misunderstood by the audience. In this trick instead of using special props, card manipulation or sleight of hand they use a mathematical concept called digital roots and taking advantage of alphabetical order.
My first inquiries were "Does this work every time?" and "Are there other sets and splits of digits this trick would work for?". To find out, I wrote a python program testing all 720 (6!) possible cases for the trick breaking 1 through 6 into two numbers with unique digits. It
uses nested loops with six variables for numbers 1 through 6. Each loop includes an if statement that only goes onto the next loop if its digit is different from all the previous ones. If it gets through the last loop, it sums the two numbers together and calculates the digital root by converting the large number into a string of characters and summing the characters as integers until a single digit remains. A frequency list with 9 positions stores the corresponding answer and prints a table at the end. Unsurprisingly all 720 cases had the number 3 as the answer.
Next, I tried all 720 combinations of three two digit numbers instead and came up with the exact same conclusion even though the sums of three smaller numbers were different from the sums of two larger numbers, the outcome was still the same. I wrote additional programs for digits 1 through 8 in sets of two four digit numbers and it returned a digital root of 9 in all 40,320 (8!) possible cases. Also digits 1 through 9 in sets of three digit numbers returning a digital root of 9 in all 362,880 (9!) cases. This trick does not only to pertain to three digit numbers with digits 1 through 6, but a lot of other combination as well!
As far as cab being in the dictionary, since the 'forced' number 3 corresponds to c, cab is always the first word listed because "c", "ca", "caa...", etc. are not valid English words. c is forced and a and b are literally the first two letters of the alphabet. What I didn't know was that this digital roots addition also works with multiplication and can be used when checking your work for minor errors when working with large numbers. Some of us have used a variation to find if numbers are divisible by 9 without realizing it!
A digital root is exactly what Penn explained how to calculate for the three or four digit number found from the sum of chosen numbers. It is calculated by adding the digits together to get a sum and repeating until one digit between 1 and 9 is remaining (ex: 856 = 1 as shown above and 1048 = 1 + 0 + 4 + 8 = 13 = 1 + 3 = 4). A very interesting mathematical pattern is that a set of whole numbers are added or multiplied, doing the same operations with their digital roots will also be equal. Below are examples using addition and multiplication (some exceptions are used to deal with negative numbers in subtraction and does not work with fractions).
Warning: Although the digital roots will always be equal, if your incorrect answer has the same digital root as the correct answer with a 1 in 9 statistical probability of happening you will miss your mistake.
dr(584) = dr(5 + 8 + 4) = dr(17) = dr(1 + 7) = 8
dr(49) = dr(4 + 9) = dr(13) = dr(1 + 3) = 4
584 + 49 = 633
dr(584) + dr(49) = dr(633) = dr(6 + 3 + 3) = dr(12) = dr(1 + 2) = 3
dr(8 + 4) = 3
584 * 49 = 28616
dr(584) * dr(49) = dr(28616) = dr(2 + 8 + 6 + 1 + 6) = dr(23) = dr(2 + 3) = 5
dr(8 * 4) = 5
I will not take time to explain the proof, but it is based on the concept that every time you "roll over" to the next place when the sum of two digits is 10 or larger, you are changing the digit sum by nine, which does not affect the digit root because the next place digit becomes one larger and the current one becomes one smaller. In the trick Penn and Teller controlled the digital root the entire time by making the guest combine the numbers 1 through 6 and as you probably figured out my now...
dr(1 + 2 + 3 + 4+ 5 + 6) = dr(21) = dr(2 + 1) = 3
...and will be the same number 3 when adding any numbers with any permutation of those digits together!
A few sources for digital roots and more generically digital sums can be found below.
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