Pi's digits with Pi's digits
Pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095...
We want every digit of Pi to be the result of a "+" and "–" operation involving the smallest number of its immediate successive single digits.
3 = -1+4 (2 digits involved)
1 = 4+1+5-9 (4 digits involved)
4 = -1+5 (2)
1 = 5+9-2-6-5 (5)
5 = 9+2-6 (3)
9 = -2+6+5 (3)
2 = -6+5+3 (3)
6 = 5-3+5+8-9 (5)
5 = -3-5-8+9+7+9-3+2-3 (9)
3 = -5-8+9+7 (4)
1 = 4+1+5-9 (4 digits involved)
4 = -1+5 (2)
1 = 5+9-2-6-5 (5)
5 = 9+2-6 (3)
9 = -2+6+5 (3)
2 = -6+5+3 (3)
6 = 5-3+5+8-9 (5)
5 = -3-5-8+9+7+9-3+2-3 (9)
3 = -5-8+9+7 (4)
...
The above 10 operations involve 40 digits; this can surely be improved.
What about the 100 best successive such operations?
Best,
É.
__________________________
Update #1 a couple of hours later
Maximilian H. was quick to correct and compute the first 100 equations:
3 = -1+4
1 = 4+1+5-9
4 = -1+5
1 = 5+9-2-6-5
5 = 9+2-6
9 = -2+6+5
2 = -6+5+3
6 = 5-3+5+8-9
5 = 3+5+8-9+7-9
3 = -5+8
5 = 8+9-7-9+3-2+3
8 = 9-7+9-3
9 = 7-9-3-2-3+8-4+6+2+6+4-3
7 = 9-3-2+3
9 = 3-2-3-8+4+6+2+6+4-3
3 = -2-3+8
2 = 3-8+4-6+2+6+4-3
3 = 8-4+6-2-6+4-3
8 = 4+6-2
4 = 6-2
6 = 2+6+4-3-3
2 = 6-4
6 = 4-3-3+8
4 = 3+3-8-3+2+7
3 = 3
3 = 8-3-2
8 = 3-2+7
3 = 2+7-9+5+0-2
2 = -7+9
7 = 9+5+0-2-8+8-4-1
9 = -5+0-2+8+8
5 = 0+2-8+8+4-1
0 = -2+8+8-4-1-9
2 = 8+8-4-1-9
8 = 8
8 = 4+1+9-7+1
4 = 1+9-7+1
1 = 9-7-1
9 = 7-1-6+9
7 = 1+6
1 = 6-9+3+9-9+3-7+5
6 = 9-3
9 = 3+9-9+3+7-5+1
3 = 9-9+3
9 = 9
9 = -3+7+5
3 = 7-5+1
7 = 5-1+0-5+8
5 = 1+0+5-8-2+0+9
1 = 0-5+8-2
0 = 5+8+2+0-9+7-4-9
5 = -8+2+0+9+7+4-9
8 = 2+0+9-7+4
2 = 0+9-7
0 = 9-7+4-9+4+4-5
9 = 7-4+9-4-4+5
7 = 4-9+4+4-5+9
4 = 9-4+4-5
9 = 4-4+5+9-2-3
4 = 4
4 = -5+9
5 = 9-2-3+0-7+8
9 = -2-3+0+7+8-1
2 = 3+0+7-8
3 = 0+7+8+1+6+4+0-6-2-8-6-2+0-8+9
0 = 7-8+1
7 = 8-1
8 = 1-6+4+0-6+2+8+6-2-0-8+9
1 = 6-4+0-6+2+8-6+2+0+8-9
6 = 4+0+6-2-8+6
4 = 0+6-2
0 = 6+2-8
6 = -2+8
2 = 8-6
8 = 6+2
6 = -2+0+8
2 = 0-8+9+9-8
0 = 8-9+9-8
8 = 9-9+8
9 = 9
9 = 8-6+2+8+0-3
8 = 6+2
6 = -2+8
2 = 8+0+3-4-8-2+5
8 = 0+3+4+8-2-5
0 = 3-4+8-2-5
3 = 4-8+2+5
4 = 8-2-5+3
8 = 2+5-3+4
2 = 5-3
5 = 3+4-2
3 = 4-2+1
4 = 2+1+1
2 = 1+1
1 = 1
1 = 7+0-6
7 = 0-6-7+9+8+2+1
0 = 6-7+9-8
6 = 7-9+8
7 = 9-8+2-1+4-8+0+8+6-5
...
Ce commentaire a été supprimé par l'auteur.
RépondreSupprimeroops, my program forgot to add a "+" (or "-") in front of the 0's, Sorry, but I think one can easily guess!
SupprimerPS2: as you can see, there are shorter solutions than in your main text in particular for the 9th and 10th digit :
Supprimer5 = -3-5-8+9+7+9-3+2-3 (9) shorter: 5 = +3+5+8-9+7-9 (6)
3 = -5-8+9+7 (4) shorter: 3 = -5+8 (2 -- and obvious!! ;-))
Génial -- merci Maximilian !
SupprimerCe commentaire a été supprimé par l'auteur.
RépondreSupprimerCe commentaire a été supprimé par l'auteur.
RépondreSupprimerCe commentaire a été supprimé par l'auteur.
Supprimeranother try :
RépondreSupprimer3 = -1+4
1 = +4+1+5-9
4 = -1+5
1 = -5+9-2-6+5
5 = +9+2-6
9 = -2+6+5
2 = -6+5+3
6 = +5-3+5+8-9
5 = +3+5+8-9+7-9
3 = -5+8
5 = -8+9-7+9+3+2-3
8 = +9-7+9-3
9 = -7-9-3-2+3+8+4+6+2+6+4-3
7 = +9-3-2+3
9 = -3-2+3-8+4+6+2+6+4-3
3 = -2-3+8
2 = +3-8+4-6+2+6+4-3
3 = -8-4+6+2+6+4-3
8 = +4+6-2
4 = +6-2
6 = +2+6+4-3-3
2 = +6-4
6 = +4-3-3+8
4 = -3-3+8-3-2+7
3 = +3
3 = +8-3-2
8 = +3-2+7
3 = -2-7+9+5+0-2
2 = -7+9
7 = -9-5+0+2+8+8+4-1
9 = -5+0-2+8+8
5 = +0+2-8+8+4-1
0 = -2+8+8-4-1-9 ********* ZERO !
2 = +8+8-4-1-9
8 = +8
8 = +4+1+9-7+1
4 = +1+9-7+1
1 = +9-7-1
9 = +7-1-6+9
7 = +1+6
1 = -6+9+3+9-9-3-7+5
6 = +9-3
9 = -3+9+9-3-7+5-1
3 = +9-9+3
9 = +9
9 = -3+7+5
3 = +7-5+1
7 = +5-1+0-5+8
5 = -1+0+5+8+2+0-9
1 = +0-5+8-2
0 = -5-8+2+0+9+7+4-9
5 = -8+2+0+9+7+4-9
8 = +2+0+9-7+4
2 = +0+9-7
0 = -9-7+4+9+4+4-5
9 = -7+4+9+4+4-5
7 = +4-9+4+4-5+9
4 = +9-4+4-5
9 = -4+4+5+9-2-3
4 = +4
4 = -5+9
5 = +9-2-3+0-7+8
9 = -2-3+0+7+8-1
2 = +3+0+7-8
3 = +0+7+8+1+6+4+0-6-2-8-6-2+0-8+9
0 = -7+8-1
7 = +8-1
8 = -1-6+4+0-6+2+8+6+2+0+8-9
1 = -6-4+0-6+2+8+6+2+0+8-9
6 = -4+0+6+2+8-6
4 = +0+6-2
0 = +6+2-8
6 = -2+8
2 = +8-6
8 = +6+2
6 = -2+0+8
2 = +0-8+9+9-8
0 = +8-9+9-8
8 = +9-9+8
9 = +9
9 = +8-6+2+8+0-3
8 = +6+2
6 = -2+8
2 = +8+0+3-4-8-2+5
8 = +0+3+4+8-2-5
0 = +3-4+8-2-5
3 = +4-8+2+5
4 = +8-2-5+3
8 = +2+5-3+4
2 = +5-3
5 = +3+4-2
3 = +4-2+1
4 = +2+1+1
2 = +1+1
1 = +1
1 = +7+0-6
7 = +0-6-7+9+8+2+1
0 = +6-7+9-8
6 = +7-9+8
7 = -9-8+2+1+4+8+0+8+6-5