The fingers game is quite simple, and can be played anywhere with another person.

1. The two players start by holding up two fingers each, one on each hand.

2. On a player's turn, that player can point with one hand to any of the three hands around him (his own or the other players'). The hand that gets pointed to adds the number of fingers on the hand that did the pointing to his own, and sticks that many fingers up. For instance, if my right hand has two fingers up and I point it to your left hand, which has one finger up, you would now show three fingers on your left hand.

3. If five fingers or more should be up, make a fist. A fist cannot be used to point with.

4. If you have one fist and your other hand has an even number of fingers, you can point with the second hand to the fist on your turn, and divide the fingers evenly between the hands. This is the only time a fist may be pointed to.

Players alternate turns, and the first player to show two fists loses.

Is there a winning strategy for either player? Which player, if either, has a better chance of winning?

Let's use a notation in which wxyz are four digits, representing a state in which the player who is about to take his turn has w fingers up in the hand with more fingers up and x fingers up in the hand with fewer fingers up (or w and x are equal if he has the same number of fingers up in each hand), and his opponent has y and z fingers up (again high-to-low count).  A fist is represented by 0 (zero).  The situation of a loss is 00yz. It will never get to wx00, as the game would have ended when the opponent had two fists.

The table below shows all the 00yz states as a loss and all the xy00 states as non-existent. The two digits to the left are the current player's hands, and the column heads refer to the opponent's hands.  It doesn't matter which hand is left or right.

This table also includes the first stage of analysis, in which state 1040 (row 10, column 40) is marked as a win for the current player, as he can point his high hand (containing 1 upraised finger) to the other player's (2nd person, or 2), high hand, making that a second fist. On the other hand, state 1033 is marked as a loss, as the only move he has available is to point his finger at one of the opponent's 3-finger hands to make the situation on the opponent's turn 4310, which is a win for that opponent, and therefore a loss for the current player.

    00   10   11   20   21   22   30   31   32   33   40   41   42   43   44
00  x    l    l    l    l    l    l    l    l    l    l    l    l    l    l
10  x    .    .    .    .    .    l    .    .    l    wh2h .    .    .    l
11  x    .    .    .    .    .    .    .    .    .    wh2h .    .    .    .
20  x    .    .    .    .    .    wh2h .    .    .    wh2h .    .    .    .
21  x    .    .    .    .    .    wh2h .    .    .    wh2h .    .    .    .
22  x    .    .    .    .    .    wh2h .    .    .    wh2h .    .    .    .
30  x    l    l    wh2h .    l    wh2h .    l    l    wh2h .    l    l    l
31  x    .    .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
32  x    .    .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
33  x    .    .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
40  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
41  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
42  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
43  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
44  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .

Now you can look for secondary levels of win/loss:

    00   10   11   20   21   22   30   31   32   33   40   41   42   43   44
00  x    l    l    l    l    l    l    l    l    l    l    l    l    l    l
10  x    .    .    wh2h .    l    l    l    l    l    wh2h .    .    wh2h l
11  x    .    .    wh2h .    .    .    .    .    .    wh2h .    .    wh2h .
20  x    .    .    .    .    .    wh2h .    .    .    wh2h .    .    .    l
21  x    .    .    .    .    .    wh2h .    .    .    wh2h .    .    .    .
22  x    wh2h .    .    .    .    wh2h .    .    wh2h wh2h .    .    wh2h .
30  x    l    l    wh2h wh2h l    wh2h wh2h l    l    wh2h wh2h l    l    l
31  x    .    .    wh2h .    .    wh2h .    .    l    wh2h .    .    .    .
32  x    wh1l .    wh2h .    .    wh2h .    wh2l wh2h wh2h .    .    wh2h .
33  x    wh1l .    wh2h wh2h .    wh2h wh2h wh2l wh2h wh2h wh2h .    wh2h .
40  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
41  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
42  x    wh2h .    wh2h .    .    wh2h wh2l wh2l wh2h wh2h .    .    wh2h .
43  x    wh2h .    wh2h .    .    wh2h wh2l wh2l wh2h wh2h .    .    wh2h .
44  x    wh2h wh2h wh2h wh2h .    wh2h wh2h wh2l wh2h wh2h wh2h .    wh2h .

where, say, 1020 allows the player a win by pointing to the opponent's high hand (2), giving that opponent a situation of 3010, already identified as a losing situation. And 1031 is a newly discovered loss in that it leads to either 4110 (already known to be a win for the opponent) or 3210 (newly found to be a win for the opponent).  (Remember, the second two digits of the state swap with the first two, as the turn changes, as the first two are always for the current player.)

Eventually, no new wins or losses are found and the table looks like this:

    00   10   11   20   21   22   30   31   32   33   40   41   42   43   44
00  x    l    l    l    l    l    l    l    l    l    l    l    l    l    l
10  x    l    l    wh2h .    l    l    l    l    l    wh2h wh2h l    wh2h l
11  x    .    .    wh2h .    .    .    .    .    .    wh2h wh2h .    wh2h .
20  x    wh1l .    .    .    .    wh2h .    .    .    wh2h .    .    .    l
21  x    wh1l .    .    .    .    wh2h .    .    .    wh2h .    .    .    .
22  x    wh2h .    .    .    .    wh2h wh2h .    wh2h wh2h wh2h .    wh2h .
30  x    l    l    wh2h wh2h l    wh2h wh2h l    l    wh2h wh2h l    l    l
31  x    .    .    wh2h wh2h .    wh2h wh2h .    l    wh2h wh2h .    .    .
32  x    wh1l .    wh2h wh2h .    wh2h wh2h wh2l wh2h wh2h wh2h .    wh2h .
33  x    wh1l .    wh2h wh2h .    wh2h wh2h wh2l wh2h wh2h wh2h .    wh2h .
40  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    l
41  x    wh2h .    wh2h .    .    wh2h .    .    .    wh2h .    .    .    .
42  x    wh2h wh2h wh2h wh2h .    wh2h wh2l wh2l wh2h wh2h wh2h .    wh2h .
43  x    wh2h .    wh2h .    .    wh2h wh2l wh2l wh2h wh2h .    .    wh2h .
44  x    wh2h wh2h wh2h wh2h wh2h wh2h wh2h wh2l wh2h wh2h wh2h wh2h wh2h wh2h

But from this table we can see that the initial position 1111 is neither a win nor a loss.  Two rational players with full information will always produce an endless cycle of plays.

To see what will happen, first let's further define our notation for moves:
h1l: player points his high hand toward his own (1st person) low hand.
l1h: player points his low hand toward his own (1st person) high hand.
h2l: player points his high hand toward his opponent's (2nd person) low hand.
h2h: player points his high hand toward his opponent's (2nd person) high hand.
l2l: player points his low hand toward his opponent's (2nd person) low hand.
l2h: player points his low hand toward his opponent's (2nd person) high hand.

In the above table this is prefaced by w to indicate he wins by doing the shown move. (There may be other winning moves, but that's an aside from the main consideration.)

The situations are laid out linearly below, with possible moves:

0000  x 
0010  l 
0011  l 
0020  l 
0021  l 
0022  l 
0030  l 
0031  l 
0032  l 
0033  l 
0040  l 
0041  l 
0042  l 
0043  l 
0044  l 
1000  x 
1010  l h2h 2010 
1011  l h2h 2110 h2l 2110 
1020  w h2h 3010
1021   h2h 3110+  h2l 2210* 
1022  l h2h 3210 h2l 3210 
1030  l h2h 4010 
1031  l h2h 4110 h2l 3210 
1032  l h2h 4210 h2l 3310 
1033  l h2h 4310 h2l 4310 
1040  w h2h 0010
1041  w h2h 1010
1042  l h2h 2010 h2l 4310 
1043  w h2h 3010
1044  l h2h 4010 h2l 4010 
1100  x h1l 0021 l1h 0021 
1110   h1l 1021+  h2h 2011+  l1h 1021+  l2h 2011+ 
1111   h1l 1121+  h2h 2111+  h2l 2111+  l1h 1121+  l2h 2111+  l2l 2111+ 
1120  w h2h 3011
1121   h1l 2121+  h2h 3111+  h2l 2211+  l1h 2121+  l2h 3111+  l2l 2211+ 
1122   h1l 2221+  h2h 3211+  h2l 3211+  l1h 2221+  l2h 3211+  l2l 3211+ 
1130   h1l 3021*  h2h 4011+  l1h 3021*  l2h 4011+ 
1131   h1l 3121*  h2h 4111+  h2l 3211+  l1h 3121*  l2h 4111+  l2l 3211+ 
1132   h1l 3221*  h2h 4211*  h2l 3311+  l1h 3221*  l2h 4211*  l2l 3311+ 
1133   h1l 3321*  h2h 4311+  h2l 4311+  l1h 3321*  l2h 4311+  l2l 4311+ 
1140  w h2h 0011
1141  w h2h 1011
1142   h1l 4221*  h2h 2011+  h2l 4311+  l1h 4221*  l2h 2011+  l2l 4311+ 
1143  w h2h 3011
1144   h1l 4421*  h2h 4011+  h2l 4011+  l1h 4421*  l2h 4011+  l2l 4011+ 
2000  x h1l 0011 
2010  w h1l 1011
2011   h1l 1111+  h2h 3120*  h2l 3120* 
2020   h1l 2011+  h2h 4020* 
2021   h1l 2111+  h2h 4120*  h2l 3220* 
2022   h1l 2211+  h2h 4220*  h2l 4220* 
2030  w h2h 0020
2031   h1l 3111+  h2h 1020*  h2l 3320* 
2032   h1l 3211+  h2h 2020+  h2l 4320* 
2033   h1l 3311+  h2h 3020*  h2l 3020* 
2040  w h2h 0020
2041   h1l 4111+  h2h 1020*  h2l 4320* 
2042   h1l 4211*  h2h 2020+  h2l 4420* 
2043   h1l 4311+  h2h 3020*  h2l 4020* 
2044  l h1l 4411 h2h 4020 h2l 4020 
2100  x h1l 0032 l1h 0031 
2110  w h1l 1032
2111   h1l 1132+  h2h 3121*  h2l 3121*  l1h 1131+  l2h 2121+  l2l 2121+ 
2120   h1l 2032+  h2h 4021+  l1h 2031+  l2h 3021* 
2121   h1l 2132+  h2h 4121+  h2l 3221*  l1h 2131+  l2h 3121*  l2l 2221+ 
2122   h1l 2232+  h2h 4221*  h2l 4221*  l1h 2231*  l2h 3221*  l2l 3221* 
2130  w h2h 0021
2131   h1l 3132+  h2h 1021+  h2l 3321*  l1h 3131*  l2h 4121+  l2l 3221* 
2132   h1l 3232*  h2h 2021+  h2l 4321+  l1h 3231*  l2h 4221*  l2l 3321* 
2133   h1l 3332*  h2h 3021*  h2l 3021*  l1h 3331*  l2h 4321+  l2l 4321+ 
2140  w h2h 0021
2141   h1l 4132+  h2h 1021+  h2l 4321+  l1h 4131+  l2h 1021+  l2l 4221* 
2142   h1l 4232*  h2h 2021+  h2l 4421*  l1h 4231*  l2h 2021+  l2l 4321+ 
2143   h1l 4332*  h2h 3021*  h2l 4021+  l1h 4331*  l2h 3021*  l2l 4421* 
2144   h1l 4432*  h2h 4021+  h2l 4021+  l1h 4431*  l2h 4021+  l2l 4021+ 
2200  x h1l 0042 l1h 0042 
2210  w h2h 3022
2211   h1l 1142+  h2h 3122+  h2l 3122+  l1h 1142+  l2h 3122+  l2l 3122+ 
2220   h1l 2042+  h2h 4022+  l1h 2042+  l2h 4022+ 
2221   h1l 2142+  h2h 4122+  h2l 3222+  l1h 2142+  l2h 4122+  l2l 3222+ 
2222   h1l 2242+  h2h 4222+  h2l 4222+  l1h 2242+  l2h 4222+  l2l 4222+ 
2230  w h2h 0022
2231  w h2h 1022
2232   h1l 3242+  h2h 2022+  h2l 4322+  l1h 3242+  l2h 2022+  l2l 4322+ 
2233  w h2h 3022
2240  w h2h 0022
2241  w h2h 1022
2242   h1l 4242+  h2h 2022+  h2l 4422*  l1h 4242+  l2h 2022+  l2l 4422* 
2243  w h2h 3022
2244   h1l 4442*  h2h 4022+  h2l 4022+  l1h 4442*  l2h 4022+  l2l 4022+ 
3000  x 
3010  l h2h 4030 
3011  l h2h 4130 h2l 4130 
3020  w h2h 0030
3021  w h2h 1030
3022  l h2h 2030 h2l 2030 
3030  w h2h 0030
3031  w h2h 1030
3032  l h2h 2030 h2l 3030 
3033  l h2h 3030 h2l 3030 
3040  w h2h 0030
3041  w h2h 1030
3042  l h2h 2030 h2l 4030 
3043  l h2h 3030 h2l 4030 
3044  l h2h 4030 h2l 4030 
3100  x h1l 0043 l1h 0041 
3110   h1l 1043*  h2h 4031+  l1h 1041*  l2h 2031+ 
3111   h1l 1143*  h2h 4131+  h2l 4131+  l1h 1141*  l2h 2131+  l2l 2131+ 
3120  w h2h 0031
3121  w h2h 1031
3122   h1l 2243*  h2h 2031+  h2l 2031+  l1h 2241*  l2h 3231*  l2l 3231* 
3130  w h2h 0031
3131  w h2h 1031
3132   h1l 3243*  h2h 2031+  h2l 3031*  l1h 3241*  l2h 4231*  l2l 3331* 
3133  l h1l 3343 h2h 3031 h2l 3031 l1h 3341 l2h 4331 l2l 4331 
3140  w h2h 0031
3141  w h2h 1031
3142   h1l 4243*  h2h 2031+  h2l 4031+  l1h 4241*  l2h 2031+  l2l 4331* 
3143   h1l 4343*  h2h 3031*  h2l 4031+  l1h 4341+  l2h 3031*  l2l 4431* 
3144   h1l 4443*  h2h 4031+  h2l 4031+  l1h 4441*  l2h 4031+  l2l 4031+ 
3200  x h1l 0030 l1h 0020 
3210  w h1l 1030
3211   h1l 1130+  h2h 4132+  h2l 4132+  l1h 1120*  l2h 3132+  l2l 3132+ 
3220  w h2h 0032
3221  w h2h 1032
3222   h1l 2230*  h2h 2032+  h2l 2032+  l1h 2220+  l2h 4232*  l2l 4232* 
3230  w h2h 0032
3231  w h2h 1032
3232  w h2l 3032
3233  w h2h 3032
3240  w h2h 0032
3241  w h2h 1032
3242   h1l 4230*  h2h 2032+  h2l 4032+  l1h 4220*  l2h 2032+  l2l 4432* 
3243  w h2h 3032
3244   h1l 4430*  h2h 4032+  h2l 4032+  l1h 4420*  l2h 4032+  l2l 4032+ 
3300  x h1l 0030 l1h 0030 
3310  w h1l 1030
3311   h1l 1130+  h2h 4133+  h2l 4133+  l1h 1130+  l2h 4133+  l2l 4133+ 
3320  w h2h 0033
3321  w h2h 1033
3322   h1l 2230*  h2h 2033+  h2l 2033+  l1h 2230*  l2h 2033+  l2l 2033+ 
3330  w h2h 0033
3331  w h2h 1033
3332  w h2l 3033
3333  w h2h 3033
3340  w h2h 0033
3341  w h2h 1033
3342   h1l 4230*  h2h 2033+  h2l 4033+  l1h 4230*  l2h 2033+  l2l 4033+ 
3343  w h2h 3033
3344   h1l 4430*  h2h 4033+  h2l 4033+  l1h 4430*  l2h 4033+  l2l 4033+ 
4000  x h1l 0022 
4010  w h2h 0040
4011   h1l 1122+  h2h 1040*  h2l 1040* 
4020  w h2h 0040
4021   h1l 2122+  h2h 1040*  h2l 2040* 
4022   h1l 2222+  h2h 2040*  h2l 2040* 
4030  w h2h 0040
4031   h1l 3122+  h2h 1040*  h2l 3040* 
4032   h1l 3222+  h2h 2040*  h2l 3040* 
4033   h1l 3322+  h2h 3040*  h2l 3040* 
4040  w h2h 0040
4041   h1l 4122+  h2h 1040*  h2l 4040* 
4042   h1l 4222+  h2h 2040*  h2l 4040* 
4043   h1l 4322+  h2h 3040*  h2l 4040* 
4044  l h1l 4422 h2h 4040 h2l 4040 
4100  x h1l 0040 l1h 0010 
4110  w h2h 0041
4111   h1l 1140*  h2h 1041*  h2l 1041*  l1h 1110+  l2h 2141+  l2l 2141+ 
4120  w h2h 0041
4121   h1l 2140*  h2h 1041*  h2l 2041+  l1h 2110*  l2h 3141*  l2l 2241* 
4122   h1l 2240*  h2h 2041+  h2l 2041+  l1h 2210*  l2h 3241*  l2l 3241* 
4130  w h2h 0041
4131   h1l 3140*  h2h 1041*  h2l 3041*  l1h 3110+  l2h 4141+  l2l 3241* 
4132   h1l 3240*  h2h 2041+  h2l 3041*  l1h 3210*  l2h 4241*  l2l 3341* 
4133   h1l 3340*  h2h 3041*  h2l 3041*  l1h 3310*  l2h 4341+  l2l 4341+ 
4140  w h2h 0041
4141   h1l 4140*  h2h 1041*  h2l 4041+  l1h 4110*  l2h 1041*  l2l 4241* 
4142   h1l 4240*  h2h 2041+  h2l 4041+  l1h 4210*  l2h 2041+  l2l 4341+ 
4143   h1l 4340*  h2h 3041*  h2l 4041+  l1h 4310*  l2h 3041*  l2l 4441* 
4144   h1l 4440*  h2h 4041+  h2l 4041+  l1h 4410*  l2h 4041+  l2l 4041+ 
4200  x h1l 0040 l1h 0020 
4210  w h2h 0042
4211  w h2h 1042
4220  w h2h 0042
4221  w h2h 1042
4222   h1l 2240*  h2h 2042+  h2l 2042+  l1h 2220+  l2h 4242+  l2l 4242+ 
4230  w h2h 0042
4231  w h2l 3042
4232  w h2l 3042
4233  w h2h 3042
4240  w h2h 0042
4241  w h2h 1042
4242   h1l 4240*  h2h 2042+  h2l 4042+  l1h 4220*  l2h 2042+  l2l 4442* 
4243  w h2h 3042
4244   h1l 4440*  h2h 4042+  h2l 4042+  l1h 4420*  l2h 4042+  l2l 4042+ 
4300  x h1l 0040 l1h 0030 
4310  w h2h 0043
4311   h1l 1140*  h2h 1043*  h2l 1043*  l1h 1130+  l2h 4143+  l2l 4143+ 
4320  w h2h 0043
4321   h1l 2140*  h2h 1043*  h2l 2043+  l1h 2130*  l2h 1043*  l2l 4243* 
4322   h1l 2240*  h2h 2043+  h2l 2043+  l1h 2230*  l2h 2043+  l2l 2043+ 
4330  w h2h 0043
4331  w h2l 3043
4332  w h2l 3043
4333  w h2h 3043
4340  w h2h 0043
4341   h1l 4140*  h2h 1043*  h2l 4043+  l1h 4130*  l2h 1043*  l2l 4443* 
4342   h1l 4240*  h2h 2043+  h2l 4043+  l1h 4230*  l2h 2043+  l2l 4043+ 
4343  w h2h 3043
4344   h1l 4440*  h2h 4043+  h2l 4043+  l1h 4430*  l2h 4043+  l2l 4043+ 
4400  x h1l 0040 l1h 0040 
4410  w h2h 0044
4411  w h2h 1044
4420  w h2h 0044
4421  w h2h 1044
4422  w h2h 2044
4430  w h2h 0044
4431  w h2h 1044
4432  w h2l 3044
4433  w h2h 3044
4440  w h2h 0044
4441  w h2h 1044
4442  w h2h 2044
4443  w h2h 3044
4444  w h2h 4044

Again, wins are shown with only one of possibly several winning moves. Losses are shown with all of their possible moves, which all lead to wins for the opponent.  Non-win/non-losses are shown with all the possible moves, with markings: + indicates it leads to a further non-win/non-loss state (part of an endless loop), while * indicates a move that will lead to a win for the opponent, so a rational player will avoid those.

Taking the starting position, 1111, all the possible moves lead to 2111 for the opponent, who then can lose the game by pointing his high hand (the one with the 2) at either of his opponent's (the original player's) hands, as 3121 is a winning state for the opponent. But the rational player with 21 against the opponent's 11, will point with one hand to the other of his own or point his low hand to one of the opponent's hands.

In each instance, in a non-win/non-loss state, the possibilities include continuance of the loop, and possibly one or more losing moves, as, if there were any winning moves, the state itself would be a winner and if all the moves were losing moves, the state would be a losing state.

So rational players in the game as presented with one finger shown on each hand to begin, will always keep the game going forever.

 

Posted by Charlie on 2004-07-11 12:48:06