Това са ВСИЧКИ решения. За краткост, премахнах различните пермутации на операндите при действията + и *, както и различните тъждествени асоциации. Признавам си без бой - с програма беше... (то и иначе всеки ви познал)=

Вижда се, че решения има за 27 четворки, като най-голямата е 58,59,60,61.


1,2,3,4:

(1+(2*3))*4


2,3,4,5:

((2*5)-3)*4


3,4,5,6:

4*(5+(6/3))


4,5,6,7:

(4*(6-5))*7
(4/(6-5))*7
4/((6-5)/7)
7/((6-5)/4)


5,6,7,8:

(6+8)*(7-5)


6,7,8,9:

(6+8)*(9-7)


7,8,9,10:

8-((7-9)*10)
8+((9-7)*10)


8,9,10,11:

8-((9-11)*10)
8+(10*(11-9))


9,10,11,12:

((10+11)/9)*12
(10+11)/(9/12)
12/(9/(10+11))


11,12,13,14:

((11+13)/12)*14
(11+13)/(12/14)
((12-11)+13)+14
(12-(11-13))+14
12-(11-(13+14))
12-((11-13)-14)
(12-(11-14))+13
12-((11-14)-13)
(13-(11-12))+14
13-(11-(12+14))
13-((11-12)-14)
14-(11-(12+13))
14-((11-12)-13)
14/(12/(11+13))


12,13,14,15:

((12-13)+14)+15
(12-(13-14))+15
12-(13-(14+15))
12-((13-14)-15)
(12-(13-15))+14
12-((13-15)-14)
(14-(13-12))+15
14-(13-(12+15))
14-((13-12)-15)
14-((13-15)-12)
15-(13-(12+14))
15-((13-12)-14)
15-((13-14)-12)


13,14,15,16:

((13+14)-15)+16
(13+14)-(15-16)
(13-(15-14))+16
13-(15-(14+16))
13-((15-14)-16)
13-((15-16)-14)
(14-(15-13))+16
14-(15-(13+16))
14-((15-13)-16)
14-((15-16)-13)
16-(15-(13+14))
16-((15-13)-14)
16-((15-14)-13)


14,15,16,17:

((14+15)+16)-17
(14+15)-(17-16)
(14*(15+17))/16
14/(16/(15+17))
(14+16)-(17-15)
14-(17-(15+16))
14-((17-15)-16)
14-((17-16)-15)
(15+16)-(17-14)
15-(17-(14+16))
15-((17-14)-16)
(15+17)/(16/14)
15-((17-16)-14)
16-(17-(14+15))
16-((17-14)-15)
16-((17-15)-14)


17,18,19,20:

18-(20/(17-19))
18+(20/(19-17))


18,19,20,21:

18-(20/(19-21))
18+(20/(21-19))


23,24,25,26:

((23+25)/24)+26
((25-23)*26)-24


24,25,26,27:

(26*(27-25))-24


26,27,28,29:

26+((27+29)/28)
(27+29)/(28-26)


27,28,29,30:

(27+29)/(30-28)
30-((27+29)/28)


29,30,31,32:

(30*(31-29))-32


30,31,32,33:

30-((31+33)/32)
(30*(33-31))-32


51,52,53,54:

(52/(51-53))+54
54-(52/(53-51))


52,53,54,55:

(52/(53-55))+54
54-(52/(55-53))


53,54,55,56:

(54/(53+55))*56
54/((53+55)/56)
56/((53+55)/54)


56,57,58,59:

(56/(57+59))*58
56/((57+59)/58)
58/((57+59)/56)


57,58,59,60:

58+(60/(57-59))
58-(60/(59-57))


58,59,60,61:

58+(60/(59-61))
58-(60/(61-59))


Интересно е да се види какво става, ако резултатът не трябва да е 28, а друго число. За 1-4 задачата е тривиална, а за резултатите до 99 се получават данните, посочени в таблицата. found показва общия брой четворки, с които може да се напише нужния израз, а maxn е най-голямата стойност на най-малкото от четирите числа, с които това е възможно. Вижда се, че maxn=(result+1)*2. Интересно е, че found сякаш няма тенденция към увеличаване с result. Решението за maxn е аналогично на случая с result=28, макар и да ми изглежда трудно да докажа, че това е най-голямата възможна стойност за всички result:

result=5 found=10 maxn=12
result=6 found=11 maxn=14
result=7 found=13 maxn=16
result=8 found=14 maxn=18
result=9 found=16 maxn=20
result=10 found=16 maxn=22
result=11 found=20 maxn=24
result=12 found=18 maxn=26
result=13 found=22 maxn=28
result=14 found=20 maxn=30
result=15 found=22 maxn=32
result=16 found=22 maxn=34
result=17 found=26 maxn=36
result=18 found=21 maxn=38
result=19 found=27 maxn=40
result=20 found=24 maxn=42
result=21 found=24 maxn=44
result=22 found=24 maxn=46
result=23 found=27 maxn=48
result=24 found=23 maxn=50
result=25 found=29 maxn=52
result=26 found=25 maxn=54
result=27 found=24 maxn=56
result=28 found=27 maxn=58
result=29 found=28 maxn=60
result=30 found=22 maxn=62
result=31 found=27 maxn=64
result=32 found=26 maxn=66
result=33 found=24 maxn=68
result=34 found=23 maxn=70
result=35 found=28 maxn=72
result=36 found=24 maxn=74
result=37 found=29 maxn=76
result=38 found=23 maxn=78
result=39 found=25 maxn=80
result=40 found=26 maxn=82
result=41 found=28 maxn=84
result=42 found=21 maxn=86
result=43 found=27 maxn=88
result=44 found=25 maxn=90
result=45 found=26 maxn=92
result=46 found=24 maxn=94
result=47 found=26 maxn=96
result=48 found=24 maxn=98
result=49 found=30 maxn=100
result=50 found=24 maxn=102
result=51 found=25 maxn=104
result=52 found=24 maxn=106
result=53 found=28 maxn=108
result=54 found=21 maxn=110
result=55 found=28 maxn=112
result=56 found=25 maxn=114
result=57 found=25 maxn=116
result=58 found=23 maxn=118
result=59 found=26 maxn=120
result=60 found=25 maxn=122
result=61 found=27 maxn=124
result=62 found=23 maxn=126
result=63 found=25 maxn=128
result=64 found=27 maxn=130
result=65 found=28 maxn=132
result=66 found=19 maxn=134
result=67 found=27 maxn=136
result=68 found=25 maxn=138
result=69 found=26 maxn=140
result=70 found=27 maxn=142
result=71 found=29 maxn=144
result=72 found=22 maxn=146
result=73 found=28 maxn=148
result=74 found=21 maxn=150
result=75 found=24 maxn=152
result=76 found=23 maxn=154
result=77 found=30 maxn=156
result=78 found=21 maxn=158
result=79 found=24 maxn=160
result=80 found=29 maxn=162
result=81 found=28 maxn=164
result=82 found=24 maxn=166
result=83 found=25 maxn=168
result=84 found=23 maxn=170
result=85 found=28 maxn=172
result=86 found=23 maxn=174
result=87 found=25 maxn=176
result=88 found=24 maxn=178
result=89 found=28 maxn=180
result=90 found=23 maxn=182
result=91 found=26 maxn=184
result=92 found=24 maxn=186
result=93 found=23 maxn=188
result=94 found=22 maxn=190
result=95 found=26 maxn=192
result=96 found=25 maxn=194
result=97 found=28 maxn=196
result=98 found=22 maxn=198
result=99 found=27 maxn=200