Since e is not zero (and neither is b), e*b is not zero, and so indeed this To take your example, the digits of base*esab are (before any carries):ī*e b*s+a*e b*a+a*s+s*e b*b+a*a+s*s+e*e a*b+s*a+e*s s*b+e*a e*bĪll of these are less or equal to b*b+a*a+s*s+e*e which is less than 10 ( iv) and since n is not a multiple of 10 there are no difficulties with ( iii) since multiplication and addition are commutative, this means that eachĭigit of the product is equal to the digit an equal distance the other side ( ii) therefore there are no carries when calculating the digits of the product Or equal to the sum of the square of the digits of n, which in turn is less ( i) any (perhaps partial) sum of pairwise products of digits of n is less than If the sum of the squares of the digits of n is less than 10 and n is notĭivisible by 10, then the product of n and the reversal of n is a palindrome. Henry Bottomley ( email) cannot quite give a proof yet, but his explanation is no less interesting. Is someone prepared to find out why that is ? A mathematical proof will be much appreciated. The basenumbers are made up of a combination of only three digits namely 0, 1 and 2 !Īlthough it is not the case, one could mistakenly believe that we are dealing with numbers in base 3 !! Some observations about Palindromic Products of Integers & their Reversals. When an integer and its reversal are unequal, their product is never a squareĮxcept when both are squares. 1089 is also known for another conjecture. The trick here is to keep using three digits even if a leading zeroĪppears in the result ! Sathiya's example thus becomes ¬Ī very sharp observation nevertheless. Struggled with what he thought is a counterexample of the above 3-digit number trick. Sathiya Subramanian ( email) from Redwood City, California It might be worthwhile to order this book as it is almost a bible for number theory enthousiasts. On the following page 64 Beiler gives a mathematical proof why this operation always works. Subtract from 371, leaving 198, reverse, giving 891, and thev Beiler, page 63, says : A common trick with an almost infinite number of variations is to have someone write a three-digit number, then write the number with the digits in reverse order, subtract the smaller from the larger, reverse the digits again and add this time you will then be able to give the answer without having received any information about the number. "Recreations in the Theory of Numbers", by Albert H. "Curious and Interesting Numbers", by David Wells, page 163, about 1089 : If a 3-digit number is reversed and the result subtracted, and that answer added to its reversal, the answer is always 1089: 623 - 326 = 297 and 297 + 792 = 1089. NO MATTER WHAT 3-DIGIT INTEGERS YOU BEGIN WITH, THE FINAL ANSWER IS ALWAYS 1089 ! If the difference is negative, then subtract its reversal. Then, if the difference is positive, add its reversal. Take any three integers from zero to nine, then subtract its reversal. This property of 1089, a truly funny number. I'm grateful to Mitch Beck for making me aware of Palindromic sum of two consecutive primes 32213 & 32233 equal to the sum of their reversiblesĮverybody must have heard of the following number trick that involves 'reverse & add'-ing If you found this article helpful let us know in the comment box.Here is a gem from Carlos Rivera's puzzle website ( Source Puzzle 52 ). So, these were the methods to type reverse 3 on your phone and PC.
Backwards 3 windows#
Note: To get a big list of symbols you need use a standard windows font like Arial or Times New Roman. Now from this list you can insert backwards 3 symbols Ɛ And many Other symbols.On the next window you can see a long list of characters and symbols.Next from the menu section Click on the “Insert” Tab.Now open any text application like MS word or Google Docs.Just follow the simple procedure mentioned below to insert the backwards 3 symbol
Backwards 3 how to#
How To Flip Mirror & Rotate Image Text In Google Docs How to Insert Backwards 3 In DocumentsĪnother simple option is there to insert the backwards 3 symbol in your writings.
Backwards 3 Pc#
First of all, Turn On your PC and open any text application like MS Word, Powerpoint, Excel Etc.
Backwards 3 code#
Follow the below given steps to make the backward 3 symbol as Ɛ using alt code :