Examples
1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321
Adding the numbers on the previous page gives 66660 and 1 + 2 + 3 + 4 = 10, so 66660 ÷ 10 = 6666.
2467 2476 2647 2674 2746 2764 4267 4276 4627 4672 4726 4762 6247 6274 6427 6472 6724 6742 7246 7264 7426 7462 7624 7642
Adding the numbers above gives 126654 and 2 + 4 + 6 + 7 = 19, so 126654 ÷ 19 = 6666.
1479 1497 1749 1794 1947 1974 4179 4197 4719 4791 4917 4971 7149 7194 7419 7491 7914 7941 9147 9174 9417 9471 9714 9741
Adding the numbers on the previous page gives 139986 and 1 + 4 + 7 + 9 = 21, so 139986 ÷ 21 = 6666.
5 Digit Numbers
- Make all the combinations using the five digits.
- Add the combinations together.
- Add the five numbers together.
- Divide ∑ the combinations divided by ∑ the five numbers and the answer will always be 266664.
Examples
12345 12354 12435 12453 12534 12543 13245 13254 13425 13452 13524 13542 14235 14253 14325 14352 14523 14532 15234 15243 15324 15342 15423 15432 21345 21354 21435 21453 21534 21543 23145 23154 23415 23451 23514 23541 24135 24153 24315 24351 24513 24531 25134 25143 25314 25341 25413 25431 31245 31254 31425 31452 31524 31542 32145 32154 32415 32451 32514 32541 34125 34152 34215 34251 34512 34521 35124 35142 35214 35241 35412 35421 41235 41253 41325 41352 41523 41532 42135 42153 42315 42351 42513 42531 43125 43152 43215 43251 43512 43521 45123 45132 45213 45231 45312 45321 51234 51243 51324 51342 51423 51432 52134 52143 52314 52341 52413 52431 53124 53142 53214 53241 53412 53421 54123 54132 54213 54231 54312 54321
Adding the above numbers gives 2999960 and 1 + 2 + 3 + 4 + 5 = 15, so 2999960 ÷ 15 = 266664.
14678 14687 14768 14786 14867 14876 16478 16487 16748 16784 16847 16874 17468 17486 17648 17684 17846 17864 18467 18476 18647 18674 18746 18764 41678 41687 41768 41786 41867 41876 46178 46187 46718 46781 46817 46871 47168 47186 47618 47681 47816 47861 48167 48176 48617 48671 48716 48761 61478 61487 61748 61784 61847 61874 64178 64187 64718 64781 64817 64871 67148 67184 67418 67481 67814 67841 68147 68174 68417 68471 68714 68741 71468 71486 71648 71684 71846 71864 74168 74186 74618 74681 74816 74861 76148 76184 76418 76481 76814 76841 78146 78164 78416 78461 78614 78641 81467 81476 81647 81674 81746 81764 84167 84176 84617 84671 84716 84761 86147 86174 86417 86471 86714 86741 87146 87164 87416 87461 87614 87641
Adding the above numbers gives 6933264 and 1 + 4 + 6 + 7 + 8 = 26, so 6933264 ÷ 26 = 26664.
6 Digit Numbers
- Make all the combinations using the six digits.
- Add the combinations together.
- Add the six numbers together.
- Divide ∑ the combinations divided by ∑ the five numbers and the answer will always be 13333320.
The Formula
(N-1)! x (10X – 1) ÷ 9
N = different amount of digits
X = number of digits
! = the factorial operation
N is how many different digits are there, e.g. 1, 2 or 3.
X is how many digits are there.
The exclamation mark the product of the integers from 1 to n, when the expression is n! From example 4!, read as four factorial is 4 x 3 x 2 x 1 = 24.
Proving The Formula
If we take the first example of the 3 digit numbers 1, 2 and 3, we can see that there are 3 digits and all the digits are different, so there are 3 types as well. If we replace N and X with 3 in the formula, the answer will be 222.
(3-1)! x (103 -1) ÷ 9 = 222
(2)! x (999) ÷ 9 = 222
2 x 999 ÷ 9 = 222
2 x 111 = 222
Four digits, e.g. 1, 2, 3 and 4 should come to 6666 and the formula will work, but this time the N and X will be replaced by 4 as there are 4 digits and they are all different.
(4-1)! x (104 -1) ÷ 9 = 6666
(3)! x (9999) ÷ 9 = 6666
6 x 9999 ÷ 9 = 6666
6 x 1111 = 6666
The answer to five digits should be 266664 and the same formula will be used. The digits can be 1, 2, 3, 4 and 5 for example, so N and X will be replaced by 5.
(5-1)! x (105 -1) ÷ 9 = 266664
(4)! x (99999) ÷ 9 = 266664
24 x 9999 ÷ 9 = 266664
24 x 11111 = 266664
Finally a six digit number should be 13333320. N and X will be replaced by 6 as there are 6 digits and all are different, e.g. 1, 2, 3, 4, 5 and 6.
(6-1)! x (106 -1) ÷ 9 = 13333320
(5)! x (999999) ÷ 9 = 13333320
120 x 999999 ÷ 9 = 13333320
120 x 111111 = 13333320
If one or two of the numbers are the same, then the answer will be different, but correct. If the numbers are 1, 1 and 2, then the combinations will be 112, 121 and 211. This means N will be replaced by 2 and X by 3.
(2-1)! x (103 -1) ÷ 9 = (1)! x (999) ÷ 9
= 1 x 999 ÷ 9
= 1 x 111
= 111
If we do it the long way, the answer will still be 111.
112 + 121 + 211 = 444
1 + 1 + 2 = 4
444 ÷ 4 = 111
If all the digits are the same, then the answer will be different again. If the digits are 1, 1 and 1, then N is replaced by 1 and X by 3.
(1-1)! x (103 -1) ÷ 9 = (0)! x (999) ÷ 9
= 1 x 999 ÷ 9
= 1 x 111
= 111
The long way is 111 + 111 + 111 = 333 and 1 + 1 + 1 = 3, so 333 ÷ 3 = 111.
0! Is defined as 1, which is a neutral element in multiplication, not multiplied by anything.
Bases
If you are given the number 13 in base ten, this means 1 ten and 3 ones. If you want to write it in base five, then it’s in number of fives and some number of ones. If the number was big enough, you might also need 25's, 125's and so on, just as in base ten you might need hundreds or thousands.
So here is 13:
___10___ 1 1 1
/ \
oooooooooo o o o
\________/ \___/
1 3
ten ones
Let's group it by 5's rather than 10's:
_5_ _5_ 1 1 1
/ \ / \
ooooo ooooo o o o
\_________/ \___/
2 3
fives ones
So we have 2 fives and 3 ones in base five, we write this as
23 (base 5).
