The Moscow Puzzles

2/12/21's puzzle 

43. A Quick Addition

(A). These six digit numbers: 328,645 491,221 816,304 117,586 671,355 508,779 183,696 882,414 can be grouped mentally and added in 8 seconds. How? (B). Say to a friend: "Write down as many four-digit numbers as you like. Then I will jot down just as many numbers and add them all up, yours and mine, in an instant." Suppose he writes: 7,621 3,057 2,794 4,518 For your first number, match his fourth number: his 4 with a 5, his 5 with a 4, his 1 with an 8, his 8 with a 1. His 4,518 plus your 5,481 equals 9,999. Match his other numbers the same way. The complete list is: 7,621 3,057 2,794 4,518 5,481 7,205 6,942 2,378 How can you know, in just a few seconds that the correct sum is 39,996? (C). Say: "Write down any two numbers. I will write a third and at once write (from left to right) the sum of the three numbers." If he writes: 72,603,294 51,273,081 what number should you write, and how do you find the total so quickly? ☟ ANSWERS ☟ (A). In the first and fifth lines, the last digits add to 10, and the other corresponding pairs of digits add to 9. The sum of the numbers is therefore 1,000,000. The second and sixth numbers, the third and seventh, and the fourth and eighth add to 1,000,000. The sum of all eight is 4,000,000. (B). To do the quick adding, you need only multiply 9,999 by 4, that is multiply 10,000 by 4 and subtract 4. (C). 48,726,918, total 172,603,293. Every digit of your third number plus the correspoinding digit of the second number equals 9. The sum is simply the first number plus 100,000,000 and minus one.