Course-18 Sum and Productpuzzle
We are given that X and Y are two integers; both X and Y are
greater than 1, X does not equal Y; and X+Y is less than 100. Foo and
Bar are two talented course-18 undergrads. Foo is given the sum, and
Bar is given the product of these numbers:
Bar says, "I
cannot find the numbers."
Foo says, "I was sure that
you could not find them."
Bar says, "Then, I found the
numbers."
Foo says, "If you could find them, then I
also found them."
What are the numbers?
We'll need a list of the prime numbers under 100:
2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
79, 83, 89, and 97
From Bar's first statement, we can assume that X*Y cannot be prime, since both X and Y are greater than 1. Nor can it be the square of a prime, since X is not equal to Y. If X*Y had exactly two proper divisors, then Bar would know the answer, so this eliminates the product of two distinct primes.
From Foo's first statement, we know that X+Y cannot be the sum of
two distinct primes. The prime sums under 100 are:
2+2=4, 2+3=5,
2+5=7, 2+7=9, 2+11=13, 2+13=15, 2+17=19, 2+19=21, 2+23=25, 2+29=31,
2+31=33, 2+37=39, 2+41=43, 2+43=45, 2+47=49, 2+53=55, 2+59=61,
2+61=63, 2+67=69, 2+71=73, 2+73=75, 2+79=81, 2+83=85, 2+89=91,
2+97=99, 3+2=5, 3+3=6, 3+5=8, 3+7=10, 3+11=14, 3+13=16, 3+17=20,
3+19=22, 3+23=26, 3+29=32, 3+31=34, 3+37=40, 3+41=44, 3+43=46,
3+47=50, 3+53=56, 3+59=62, 3+61=64, 3+67=70, 3+71=74, 3+73=76,
3+79=82, 3+83=86, 3+89=92, 5+2=7, 5+3=8, 5+5=10, 5+7=12, 5+11=16,
5+13=18, 5+17=22, 5+19=24, 5+23=28, 5+29=34, 5+31=36, 5+37=42,
5+41=46, 5+43=48, 5+47=52, 5+53=58, 5+59=64, 5+61=66, 5+67=72,
5+71=76, 5+73=78, 5+79=84, 5+83=88, 5+89=94, 7+2=9, 7+3=10, 7+5=12,
7+7=14, 7+11=18, 7+13=20, 7+17=24, 7+19=26, 7+23=30, 7+29=36, 7+31=38,
7+37=44, 7+41=48, 7+43=50, 7+47=54, 7+53=60, 7+59=66, 7+61=68,
7+67=74, 7+71=78, 7+73=80, 7+79=86, 7+83=90, 7+89=96, 11+2=13,
11+3=14, 11+5=16, 11+7=18, 11+11=22, 11+13=24, 11+17=28, 11+19=30,
11+23=34, 11+29=40, 11+31=42, 11+37=48, 11+41=52, 11+43=54, 11+47=58,
11+53=64, 11+59=70, 11+61=72, 11+67=78, 11+71=82, 11+73=84, 11+79=90,
11+83=94, 13+2=15, 13+3=16, 13+5=18, 13+7=20, 13+11=24, 13+13=26,
13+17=30, 13+19=32, 13+23=36, 13+29=42, 13+31=44, 13+37=50, 13+41=54,
13+43=56, 13+47=60, 13+53=66, 13+59=72, 13+61=74, 13+67=80, 13+71=84,
13+73=86, 13+79=92, 13+83=96, 17+2=19, 17+3=20, 17+5=22, 17+7=24,
17+11=28, 17+13=30, 17+17=34, 17+19=36, 17+23=40, 17+29=46, 17+31=48,
17+37=54, 17+41=58, 17+43=60, 17+47=64, 17+53=70, 17+59=76, 17+61=78,
17+67=84, 17+71=88, 17+73=90, 17+79=96, 19+2=21, 19+3=22, 19+5=24,
19+7=26, 19+11=30, 19+13=32, 19+17=36, 19+19=38, 19+23=42, 19+29=48,
19+31=50, 19+37=56, 19+41=60, 19+43=62, 19+47=66, 19+53=72, 19+59=78,
19+61=80, 19+67=86, 19+71=90, 19+73=92, 19+79=98, 23+2=25, 23+3=26,
23+5=28, 23+7=30, 23+11=34, 23+13=36, 23+17=40, 23+19=42, 23+23=46,
23+29=52, 23+31=54, 23+37=60, 23+41=64, 23+43=66, 23+47=70, 23+53=76,
23+59=82, 23+61=84, 23+67=90, 23+71=94, 23+73=96, 29+2=31, 29+3=32,
29+5=34, 29+7=36, 29+11=40, 29+13=42, 29+17=46, 29+19=48, 29+23=52,
29+29=58, 29+31=60, 29+37=66, 29+41=70, 29+43=72, 29+47=76, 29+53=82,
29+59=88, 29+61=90, 29+67=96, 31+2=33, 31+3=34, 31+5=36, 31+7=38,
31+11=42, 31+13=44, 31+17=48, 31+19=50, 31+23=54, 31+29=60, 31+31=62,
31+37=68, 31+41=72, 31+43=74, 31+47=78, 31+53=84, 31+59=90, 31+61=92,
31+67=98, 37+2=39, 37+3=40, 37+5=42, 37+7=44, 37+11=48, 37+13=50,
37+17=54, 37+19=56, 37+23=60, 37+29=66, 37+31=68, 37+37=74, 37+41=78,
37+43=80, 37+47=84, 37+53=90, 37+59=96, 37+61=98, 41+2=43, 41+3=44,
41+5=46, 41+7=48, 41+11=52, 41+13=54, 41+17=58, 41+19=60, 41+23=64,
41+29=70, 41+31=72, 41+37=78, 41+41=82, 41+43=84, 41+47=88, 41+53=94,
43+2=45, 43+3=46, 43+5=48, 43+7=50, 43+11=54, 43+13=56, 43+17=60,
43+19=62, 43+23=66, 43+29=72, 43+31=74, 43+37=80, 43+41=84, 43+43=86,
43+47=90, 43+53=96, 47+2=49, 47+3=50, 47+5=52, 47+7=54, 47+11=58,
47+13=60, 47+17=64, 47+19=66, 47+23=70, 47+29=76, 47+31=78, 47+37=84,
47+41=88, 47+43=90, 47+47=94, 53+2=55, 53+3=56, 53+5=58, 53+7=60,
53+11=64, 53+13=66, 53+17=70, 53+19=72, 53+23=76, 53+29=82, 53+31=84,
53+37=90, 53+41=94, 53+43=96, 59+2=61, 59+3=62, 59+5=64, 59+7=66,
59+11=70, 59+13=72, 59+17=76, 59+19=78, 59+23=82, 59+29=88, 59+31=90,
59+37=96, 61+2=63, 61+3=64, 61+5=66, 61+7=68, 61+11=72, 61+13=74,
61+17=78, 61+19=80, 61+23=84, 61+29=90, 61+31=92, 61+37=98, 67+2=69,
67+3=70, 67+5=72, 67+7=74, 67+11=78, 67+13=80, 67+17=84, 67+19=86,
67+23=90, 67+29=96, 67+31=98, 71+2=73, 71+3=74, 71+5=76, 71+7=78,
71+11=82, 71+13=84, 71+17=88, 71+19=90, 71+23=94, 73+2=75, 73+3=76,
73+5=78, 73+7=80, 73+11=84, 73+13=86, 73+17=90, 73+19=92, 73+23=96,
79+2=81, 79+3=82, 79+5=84, 79+7=86, 79+11=90, 79+13=92, 79+17=96,
79+19=98, 83+2=85, 83+3=86, 83+5=88, 83+7=90, 83+11=94, 83+13=96,
89+2=91, 89+3=92, 89+5=94, 89+7=96, and 97+2=99
The numbers less than 100 that are not prime sums are:
11,
17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79,
83, 87, 89, 93, 95, and 97
Since these sums are all odd, we know
that one of the numbers must be odd and one must be even, and from
Bar's second statement, we know that any factorization pair where the
sum is not in this list is eliminated, e.g., X*Y=12 has factors 2*6
and 3*4, but neither sums 8 or 7 are on the list.
The possible products of the two factors the sum of which are in
the list are:
2+9=11 2x9=18, 3+8=11 3x8=24, 4+7=11 4x7=28,
5+6=11 5x6=30, 2+15=17 2x15=30, 3+14=17 3x14=42, 4+13=17 4x13=52,
5+12=17 5x12=60, 6+11=17 6x11=66, 7+10=17 7x10=70, 8+9=17 8x9=72,
2+21=23 2x21=42, 3+20=23 3x20=60, 4+19=23 4x19=76, 5+18=23 5x18=90,
6+17=23 6x17=102, 7+16=23 7x16=112, 8+15=23 8x15=120, 9+14=23
9x14=126, 10+13=23 10x13=130, 11+12=23 11x12=132, 2+25=27 2x25=50,
3+24=27 3x24=72, 4+23=27 4x23=92, 5+22=27 5x22=110, 6+21=27 6x21=126,
7+20=27 7x20=140, 8+19=27 8x19=152, 9+18=27 9x18=162, 10+17=27
10x17=170, 11+16=27 11x16=176, 12+15=27 12x15=180, 13+14=27 13x14=182,
2+27=29 2x27=54, 3+26=29 3x26=78, 4+25=29 4x25=100, 5+24=29 5x24=120,
6+23=29 6x23=138, 7+22=29 7x22=154, 8+21=29 8x21=168, 9+20=29
9x20=180, 10+19=29 10x19=190, 11+18=29 11x18=198, 12+17=29 12x17=204,
13+16=29 13x16=208, 14+15=29 14x15=210, 2+33=35 2x33=66, 3+32=35
3x32=96, 4+31=35 4x31=124, 5+30=35 5x30=150, 6+29=35 6x29=174, 7+28=35
7x28=196, 8+27=35 8x27=216, 9+26=35 9x26=234, 10+25=35 10x25=250,
11+24=35 11x24=264, 12+23=35 12x23=276, 13+22=35 13x22=286, 14+21=35
14x21=294, 15+20=35 15x20=300, 16+19=35 16x19=304, 17+18=35 17x18=306,
2+35=37 2x35=70, 3+34=37 3x34=102, 4+33=37 4x33=132, 5+32=37 5x32=160,
6+31=37 6x31=186, 7+30=37 7x30=210, 8+29=37 8x29=232, 9+28=37
9x28=252, 10+27=37 10x27=270, 11+26=37 11x26=286, 12+25=37 12x25=300,
13+24=37 13x24=312, 14+23=37 14x23=322, 15+22=37 15x22=330, 16+21=37
16x21=336, 17+20=37 17x20=340, 18+19=37 18x19=342, 2+39=41 2x39=78,
3+38=41 3x38=114, 4+37=41 4x37=148, 5+36=41 5x36=180, 6+35=41
6x35=210, 7+34=41 7x34=238, 8+33=41 8x33=264, 9+32=41 9x32=288,
10+31=41 10x31=310, 11+30=41 11x30=330, 12+29=41 12x29=348, 13+28=41
13x28=364, 14+27=41 14x27=378, 15+26=41 15x26=390, 16+25=41 16x25=400,
17+24=41 17x24=408, 18+23=41 18x23=414, 19+22=41 19x22=418, 20+21=41
20x21=420, 2+45=47 2x45=90, 3+44=47 3x44=132, 4+43=47 4x43=172,
5+42=47 5x42=210, 6+41=47 6x41=246, 7+40=47 7x40=280, 8+39=47
8x39=312, 9+38=47 9x38=342, 10+37=47 10x37=370, 11+36=47 11x36=396,
12+35=47 12x35=420, 13+34=47 13x34=442, 14+33=47 14x33=462, 15+32=47
15x32=480, 16+31=47 16x31=496, 17+30=47 17x30=510, 18+29=47 18x29=522,
19+28=47 19x28=532, 20+27=47 20x27=540, 21+26=47 21x26=546, 22+25=47
22x25=550, 23+24=47 23x24=552, 2+49=51 2x49=98, 3+48=51 3x48=144,
4+47=51 4x47=188, 5+46=51 5x46=230, 6+45=51 6x45=270, 7+44=51
7x44=308, 8+43=51 8x43=344, 9+42=51 9x42=378, 10+41=51 10x41=410,
11+40=51 11x40=440, 12+39=51 12x39=468, 13+38=51 13x38=494, 14+37=51
14x37=518, 15+36=51 15x36=540, 16+35=51 16x35=560, 17+34=51 17x34=578,
18+33=51 18x33=594, 19+32=51 19x32=608, 20+31=51 20x31=620, 21+30=51
21x30=630, 22+29=51 22x29=638, 23+28=51 23x28=644, 24+27=51 24x27=648,
25+26=51 25x26=650, 2+51=53 2x51=102, 3+50=53 3x50=150, 4+49=53
4x49=196, 5+48=53 5x48=240, 6+47=53 6x47=282, 7+46=53 7x46=322,
8+45=53 8x45=360, 9+44=53 9x44=396, 10+43=53 10x43=430, 11+42=53
11x42=462, 12+41=53 12x41=492, 13+40=53 13x40=520, 14+39=53 14x39=546,
15+38=53 15x38=570, 16+37=53 16x37=592, 17+36=53 17x36=612, 18+35=53
18x35=630, 19+34=53 19x34=646, 20+33=53 20x33=660, 21+32=53 21x32=672,
22+31=53 22x31=682, 23+30=53 23x30=690, 24+29=53 24x29=696, 25+28=53
25x28=700, 26+27=53 26x27=702, 2+55=57 2x55=110, 3+54=57 3x54=162,
4+53=57 4x53=212, 5+52=57 5x52=260, 6+51=57 6x51=306, 7+50=57
7x50=350, 8+49=57 8x49=392, 9+48=57 9x48=432, 10+47=57 10x47=470,
11+46=57 11x46=506, 12+45=57 12x45=540, 13+44=57 13x44=572, 14+43=57
14x43=602, 15+42=57 15x42=630, 16+41=57 16x41=656, 17+40=57 17x40=680,
18+39=57 18x39=702, 19+38=57 19x38=722, 20+37=57 20x37=740, 21+36=57
21x36=756, 22+35=57 22x35=770, 23+34=57 23x34=782, 24+33=57 24x33=792,
25+32=57 25x32=800, 26+31=57 26x31=806, 27+30=57 27x30=810, 28+29=57
28x29=812, 2+57=59 2x57=114, 3+56=59 3x56=168, 4+55=59 4x55=220,
5+54=59 5x54=270, 6+53=59 6x53=318, 7+52=59 7x52=364, 8+51=59
8x51=408, 9+50=59 9x50=450, 10+49=59 10x49=490, 11+48=59 11x48=528,
12+47=59 12x47=564, 13+46=59 13x46=598, 14+45=59 14x45=630, 15+44=59
15x44=660, 16+43=59 16x43=688, 17+42=59 17x42=714, 18+41=59 18x41=738,
19+40=59 19x40=760, 20+39=59 20x39=780, 21+38=59 21x38=798, 22+37=59
22x37=814, 23+36=59 23x36=828, 24+35=59 24x35=840, 25+34=59 25x34=850,
26+33=59 26x33=858, 27+32=59 27x32=864, 28+31=59 28x31=868, 29+30=59
29x30=870, 2+63=65 2x63=126, 3+62=65 3x62=186, 4+61=65 4x61=244,
5+60=65 5x60=300, 6+59=65 6x59=354, 7+58=65 7x58=406, 8+57=65
8x57=456, 9+56=65 9x56=504, 10+55=65 10x55=550, 11+54=65 11x54=594,
12+53=65 12x53=636, 13+52=65 13x52=676, 14+51=65 14x51=714, 15+50=65
15x50=750, 16+49=65 16x49=784, 17+48=65 17x48=816, 18+47=65 18x47=846,
19+46=65 19x46=874, 20+45=65 20x45=900, 21+44=65 21x44=924, 22+43=65
22x43=946, 23+42=65 23x42=966, 24+41=65 24x41=984, 25+40=65
25x40=1000, 26+39=65 26x39=1014, 27+38=65 27x38=1026, 28+37=65
28x37=1036, 29+36=65 29x36=1044, 30+35=65 30x35=1050, 31+34=65
31x34=1054, 32+33=65 32x33=1056, 2+65=67 2x65=130, 3+64=67 3x64=192,
4+63=67 4x63=252, 5+62=67 5x62=310, 6+61=67 6x61=366, 7+60=67
7x60=420, 8+59=67 8x59=472, 9+58=67 9x58=522, 10+57=67 10x57=570,
11+56=67 11x56=616, 12+55=67 12x55=660, 13+54=67 13x54=702, 14+53=67
14x53=742, 15+52=67 15x52=780, 16+51=67 16x51=816, 17+50=67 17x50=850,
18+49=67 18x49=882, 19+48=67 19x48=912, 20+47=67 20x47=940, 21+46=67
21x46=966, 22+45=67 22x45=990, 23+44=67 23x44=1012, 24+43=67
24x43=1032, 25+42=67 25x42=1050, 26+41=67 26x41=1066, 27+40=67
27x40=1080, 28+39=67 28x39=1092, 29+38=67 29x38=1102, 30+37=67
30x37=1110, 31+36=67 31x36=1116, 32+35=67 32x35=1120, 33+34=67
33x34=1122, 2+69=71 2x69=138, 3+68=71 3x68=204, 4+67=71 4x67=268,
5+66=71 5x66=330, 6+65=71 6x65=390, 7+64=71 7x64=448, 8+63=71
8x63=504, 9+62=71 9x62=558, 10+61=71 10x61=610, 11+60=71 11x60=660,
12+59=71 12x59=708, 13+58=71 13x58=754, 14+57=71 14x57=798, 15+56=71
15x56=840, 16+55=71 16x55=880, 17+54=71 17x54=918, 18+53=71 18x53=954,
19+52=71 19x52=988, 20+51=71 20x51=1020, 21+50=71 21x50=1050, 22+49=71
22x49=1078, 23+48=71 23x48=1104, 24+47=71 24x47=1128, 25+46=71
25x46=1150, 26+45=71 26x45=1170, 27+44=71 27x44=1188, 28+43=71
28x43=1204, 29+42=71 29x42=1218, 30+41=71 30x41=1230, 31+40=71
31x40=1240, 32+39=71 32x39=1248, 33+38=71 33x38=1254, 34+37=71
34x37=1258, 35+36=71 35x36=1260, 2+75=77 2x75=150, 3+74=77 3x74=222,
4+73=77 4x73=292, 5+72=77 5x72=360, 6+71=77 6x71=426, 7+70=77
7x70=490, 8+69=77 8x69=552, 9+68=77 9x68=612, 10+67=77 10x67=670,
11+66=77 11x66=726, 12+65=77 12x65=780, 13+64=77 13x64=832, 14+63=77
14x63=882, 15+62=77 15x62=930, 16+61=77 16x61=976, 17+60=77
17x60=1020, 18+59=77 18x59=1062, 19+58=77 19x58=1102, 20+57=77
20x57=1140, 21+56=77 21x56=1176, 22+55=77 22x55=1210, 23+54=77
23x54=1242, 24+53=77 24x53=1272, 25+52=77 25x52=1300, 26+51=77
26x51=1326, 27+50=77 27x50=1350, 28+49=77 28x49=1372, 29+48=77
29x48=1392, 30+47=77 30x47=1410, 31+46=77 31x46=1426, 32+45=77
32x45=1440, 33+44=77 33x44=1452, 34+43=77 34x43=1462, 35+42=77
35x42=1470, 36+41=77 36x41=1476, 37+40=77 37x40=1480, 38+39=77
38x39=1482, 2+77=79 2x77=154, 3+76=79 3x76=228, 4+75=79 4x75=300,
5+74=79 5x74=370, 6+73=79 6x73=438, 7+72=79 7x72=504, 8+71=79
8x71=568, 9+70=79 9x70=630, 10+69=79 10x69=690, 11+68=79 11x68=748,
12+67=79 12x67=804, 13+66=79 13x66=858, 14+65=79 14x65=910, 15+64=79
15x64=960, 16+63=79 16x63=1008, 17+62=79 17x62=1054, 18+61=79
18x61=1098, 19+60=79 19x60=1140, 20+59=79 20x59=1180, 21+58=79
21x58=1218, 22+57=79 22x57=1254, 23+56=79 23x56=1288, 24+55=79
24x55=1320, 25+54=79 25x54=1350, 26+53=79 26x53=1378, 27+52=79
27x52=1404, 28+51=79 28x51=1428, 29+50=79 29x50=1450, 30+49=79
30x49=1470, 31+48=79 31x48=1488, 32+47=79 32x47=1504, 33+46=79
33x46=1518, 34+45=79 34x45=1530, 35+44=79 35x44=1540, 36+43=79
36x43=1548, 37+42=79 37x42=1554, 38+41=79 38x41=1558, 39+40=79
39x40=1560, 2+81=83 2x81=162, 3+80=83 3x80=240, 4+79=83 4x79=316,
5+78=83 5x78=390, 6+77=83 6x77=462, 7+76=83 7x76=532, 8+75=83
8x75=600, 9+74=83 9x74=666, 10+73=83 10x73=730, 11+72=83 11x72=792,
12+71=83 12x71=852, 13+70=83 13x70=910, 14+69=83 14x69=966, 15+68=83
15x68=1020, 16+67=83 16x67=1072, 17+66=83 17x66=1122, 18+65=83
18x65=1170, 19+64=83 19x64=1216, 20+63=83 20x63=1260, 21+62=83
21x62=1302, 22+61=83 22x61=1342, 23+60=83 23x60=1380, 24+59=83
24x59=1416, 25+58=83 25x58=1450, 26+57=83 26x57=1482, 27+56=83
27x56=1512, 28+55=83 28x55=1540, 29+54=83 29x54=1566, 30+53=83
30x53=1590, 31+52=83 31x52=1612, 32+51=83 32x51=1632, 33+50=83
33x50=1650, 34+49=83 34x49=1666, 35+48=83 35x48=1680, 36+47=83
36x47=1692, 37+46=83 37x46=1702, 38+45=83 38x45=1710, 39+44=83
39x44=1716, 40+43=83 40x43=1720, 41+42=83 41x42=1722, 2+85=87
2x85=170, 3+84=87 3x84=252, 4+83=87 4x83=332, 5+82=87 5x82=410,
6+81=87 6x81=486, 7+80=87 7x80=560, 8+79=87 8x79=632, 9+78=87
9x78=702, 10+77=87 10x77=770, 11+76=87 11x76=836, 12+75=87 12x75=900,
13+74=87 13x74=962, 14+73=87 14x73=1022, 15+72=87 15x72=1080, 16+71=87
16x71=1136, 17+70=87 17x70=1190, 18+69=87 18x69=1242, 19+68=87
19x68=1292, 20+67=87 20x67=1340, 21+66=87 21x66=1386, 22+65=87
22x65=1430, 23+64=87 23x64=1472, 24+63=87 24x63=1512, 25+62=87
25x62=1550, 26+61=87 26x61=1586, 27+60=87 27x60=1620, 28+59=87
28x59=1652, 29+58=87 29x58=1682, 30+57=87 30x57=1710, 31+56=87
31x56=1736, 32+55=87 32x55=1760, 33+54=87 33x54=1782, 34+53=87
34x53=1802, 35+52=87 35x52=1820, 36+51=87 36x51=1836, 37+50=87
37x50=1850, 38+49=87 38x49=1862, 39+48=87 39x48=1872, 40+47=87
40x47=1880, 41+46=87 41x46=1886, 42+45=87 42x45=1890, 43+44=87
43x44=1892, 2+87=89 2x87=174, 3+86=89 3x86=258, 4+85=89 4x85=340,
5+84=89 5x84=420, 6+83=89 6x83=498, 7+82=89 7x82=574, 8+81=89
8x81=648, 9+80=89 9x80=720, 10+79=89 10x79=790, 11+78=89 11x78=858,
12+77=89 12x77=924, 13+76=89 13x76=988, 14+75=89 14x75=1050, 15+74=89
15x74=1110, 16+73=89 16x73=1168, 17+72=89 17x72=1224, 18+71=89
18x71=1278, 19+70=89 19x70=1330, 20+69=89 20x69=1380, 21+68=89
21x68=1428, 22+67=89 22x67=1474, 23+66=89 23x66=1518, 24+65=89
24x65=1560, 25+64=89 25x64=1600, 26+63=89 26x63=1638, 27+62=89
27x62=1674, 28+61=89 28x61=1708, 29+60=89 29x60=1740, 30+59=89
30x59=1770, 31+58=89 31x58=1798, 32+57=89 32x57=1824, 33+56=89
33x56=1848, 34+55=89 34x55=1870, 35+54=89 35x54=1890, 36+53=89
36x53=1908, 37+52=89 37x52=1924, 38+51=89 38x51=1938, 39+50=89
39x50=1950, 40+49=89 40x49=1960, 41+48=89 41x48=1968, 42+47=89
42x47=1974, 43+46=89 43x46=1978, 44+45=89 44x45=1980, 2+91=93
2x91=182, 3+90=93 3x90=270, 4+89=93 4x89=356, 5+88=93 5x88=440,
6+87=93 6x87=522, 7+86=93 7x86=602, 8+85=93 8x85=680, 9+84=93
9x84=756, 10+83=93 10x83=830, 11+82=93 11x82=902, 12+81=93 12x81=972,
13+80=93 13x80=1040, 14+79=93 14x79=1106, 15+78=93 15x78=1170,
16+77=93 16x77=1232, 17+76=93 17x76=1292, 18+75=93 18x75=1350,
19+74=93 19x74=1406, 20+73=93 20x73=1460, 21+72=93 21x72=1512,
22+71=93 22x71=1562, 23+70=93 23x70=1610, 24+69=93 24x69=1656,
25+68=93 25x68=1700, 26+67=93 26x67=1742, 27+66=93 27x66=1782,
28+65=93 28x65=1820, 29+64=93 29x64=1856, 30+63=93 30x63=1890,
31+62=93 31x62=1922, 32+61=93 32x61=1952, 33+60=93 33x60=1980,
34+59=93 34x59=2006, 35+58=93 35x58=2030, 36+57=93 36x57=2052,
37+56=93 37x56=2072, 38+55=93 38x55=2090, 39+54=93 39x54=2106,
40+53=93 40x53=2120, 41+52=93 41x52=2132, 42+51=93 42x51=2142,
43+50=93 43x50=2150, 44+49=93 44x49=2156, 45+48=93 45x48=2160,
46+47=93 46x47=2162, 2+93=95 2x93=186, 3+92=95 3x92=276, 4+91=95
4x91=364, 5+90=95 5x90=450, 6+89=95 6x89=534, 7+88=95 7x88=616,
8+87=95 8x87=696, 9+86=95 9x86=774, 10+85=95 10x85=850, 11+84=95
11x84=924, 12+83=95 12x83=996, 13+82=95 13x82=1066, 14+81=95
14x81=1134, 15+80=95 15x80=1200, 16+79=95 16x79=1264, 17+78=95
17x78=1326, 18+77=95 18x77=1386, 19+76=95 19x76=1444, 20+75=95
20x75=1500, 21+74=95 21x74=1554, 22+73=95 22x73=1606, 23+72=95
23x72=1656, 24+71=95 24x71=1704, 25+70=95 25x70=1750, 26+69=95
26x69=1794, 27+68=95 27x68=1836, 28+67=95 28x67=1876, 29+66=95
29x66=1914, 30+65=95 30x65=1950, 31+64=95 31x64=1984, 32+63=95
32x63=2016, 33+62=95 33x62=2046, 34+61=95 34x61=2074, 35+60=95
35x60=2100, 36+59=95 36x59=2124, 37+58=95 37x58=2146, 38+57=95
38x57=2166, 39+56=95 39x56=2184, 40+55=95 40x55=2200, 41+54=95
41x54=2214, 42+53=95 42x53=2226, 43+52=95 43x52=2236, 44+51=95
44x51=2244, 45+50=95 45x50=2250, 46+49=95 46x49=2254, 47+48=95
47x48=2256, 2+95=97 2x95=190, 3+94=97 3x94=282, 4+93=97 4x93=372,
5+92=97 5x92=460, 6+91=97 6x91=546, 7+90=97 7x90=630, 8+89=97
8x89=712, 9+88=97 9x88=792, 10+87=97 10x87=870, 11+86=97 11x86=946,
12+85=97 12x85=1020, 13+84=97 13x84=1092, 14+83=97 14x83=1162,
15+82=97 15x82=1230, 16+81=97 16x81=1296, 17+80=97 17x80=1360,
18+79=97 18x79=1422, 19+78=97 19x78=1482, 20+77=97 20x77=1540,
21+76=97 21x76=1596, 22+75=97 22x75=1650, 23+74=97 23x74=1702,
24+73=97 24x73=1752, 25+72=97 25x72=1800, 26+71=97 26x71=1846,
27+70=97 27x70=1890, 28+69=97 28x69=1932, 29+68=97 29x68=1972,
30+67=97 30x67=2010, 31+66=97 31x66=2046, 32+65=97 32x65=2080,
33+64=97 33x64=2112, 34+63=97 34x63=2142, 35+62=97 35x62=2170,
36+61=97 36x61=2196, 37+60=97 37x60=2220, 38+59=97 38x59=2242,
39+58=97 39x58=2262, 40+57=97 40x57=2280, 41+56=97 41x56=2296,
42+55=97 42x55=2310, 43+54=97 43x54=2322, 44+53=97 44x53=2332,
45+52=97 45x52=2340, 46+51=97 46x51=2346, 47+50=97 47x50=2350, and
48+49=97 48x49=2352
We also know from Bar that there has to be a unique product, e.g.,
we can eliminate X*Y=120, because 120=5*24=15*8, and 5+24=29 and
15+8=23. The products that are unique from the previous list are:
11: 18, 11: 24, 11: 28, 17: 52, 23: 76, 23: 112, 27: 50, 27: 92, 27:
140, 27: 152, 27: 176, 29: 54, 29: 100, 29: 198, 29: 208, 35: 96, 35:
124, 35: 216, 35: 234, 35: 250, 35: 294, 35: 304, 37: 160, 37: 232,
37: 336, 41: 148, 41: 238, 41: 288, 41: 348, 41: 400, 41: 414, 41:
418, 47: 172, 47: 246, 47: 280, 47: 442, 47: 480, 47: 496, 47: 510,
51: 98, 51: 144, 51: 188, 51: 230, 51: 308, 51: 344, 51: 468, 51: 494,
51: 518, 51: 578, 51: 608, 51: 620, 51: 638, 51: 644, 51: 650, 53:
430, 53: 492, 53: 520, 53: 592, 53: 646, 53: 672, 53: 682, 53: 700,
57: 212, 57: 260, 57: 350, 57: 392, 57: 432, 57: 470, 57: 506, 57:
572, 57: 656, 57: 722, 57: 740, 57: 782, 57: 800, 57: 806, 57: 810,
57: 812, 59: 220, 59: 318, 59: 528, 59: 564, 59: 598, 59: 688, 59:
738, 59: 760, 59: 814, 59: 828, 59: 864, 59: 868, 65: 244, 65: 354,
65: 406, 65: 456, 65: 636, 65: 676, 65: 750, 65: 784, 65: 846, 65:
874, 65: 984, 65: 1000, 65: 1014, 65: 1026, 65: 1036, 65: 1044, 65:
1056, 67: 192, 67: 366, 67: 472, 67: 742, 67: 912, 67: 940, 67: 990,
67: 1012, 67: 1032, 67: 1116, 67: 1120, 71: 268, 71: 448, 71: 558, 71:
610, 71: 708, 71: 754, 71: 880, 71: 918, 71: 954, 71: 1078, 71: 1104,
71: 1128, 71: 1150, 71: 1188, 71: 1204, 71: 1240, 71: 1248, 71: 1258,
77: 222, 77: 292, 77: 426, 77: 670, 77: 726, 77: 832, 77: 930, 77:
976, 77: 1062, 77: 1176, 77: 1210, 77: 1272, 77: 1300, 77: 1372, 77:
1392, 77: 1410, 77: 1426, 77: 1440, 77: 1452, 77: 1462, 77: 1476, 77:
1480, 79: 228, 79: 438, 79: 568, 79: 748, 79: 804, 79: 960, 79: 1008,
79: 1098, 79: 1180, 79: 1288, 79: 1320, 79: 1378, 79: 1404, 79: 1488,
79: 1504, 79: 1530, 79: 1548, 79: 1558, 83: 316, 83: 600, 83: 666, 83:
730, 83: 852, 83: 1072, 83: 1216, 83: 1302, 83: 1342, 83: 1416, 83:
1566, 83: 1590, 83: 1612, 83: 1632, 83: 1666, 83: 1680, 83: 1692, 83:
1716, 83: 1720, 83: 1722, 87: 332, 87: 486, 87: 632, 87: 836, 87: 962,
87: 1022, 87: 1136, 87: 1190, 87: 1340, 87: 1430, 87: 1472, 87: 1550,
87: 1586, 87: 1620, 87: 1652, 87: 1682, 87: 1736, 87: 1760, 87: 1802,
87: 1850, 87: 1862, 87: 1872, 87: 1880, 87: 1886, 87: 1892, 89: 258,
89: 498, 89: 574, 89: 720, 89: 790, 89: 1168, 89: 1224, 89: 1278, 89:
1330, 89: 1474, 89: 1600, 89: 1638, 89: 1674, 89: 1708, 89: 1740, 89:
1770, 89: 1798, 89: 1824, 89: 1848, 89: 1870, 89: 1908, 89: 1924, 89:
1938, 89: 1960, 89: 1968, 89: 1974, 89: 1978, 93: 356, 93: 830, 93:
902, 93: 972, 93: 1040, 93: 1106, 93: 1232, 93: 1406, 93: 1460, 93:
1562, 93: 1610, 93: 1700, 93: 1742, 93: 1856, 93: 1922, 93: 1952, 93:
2006, 93: 2030, 93: 2052, 93: 2072, 93: 2090, 93: 2106, 93: 2120, 93:
2132, 93: 2150, 93: 2156, 93: 2160, 93: 2162, 95: 534, 95: 774, 95:
996, 95: 1134, 95: 1200, 95: 1264, 95: 1444, 95: 1500, 95: 1606, 95:
1704, 95: 1750, 95: 1794, 95: 1876, 95: 1914, 95: 1984, 95: 2016, 95:
2074, 95: 2100, 95: 2124, 95: 2146, 95: 2166, 95: 2184, 95: 2200, 95:
2214, 95: 2226, 95: 2236, 95: 2244, 95: 2250, 95: 2254, 95: 2256, 97:
372, 97: 460, 97: 712, 97: 1162, 97: 1296, 97: 1360, 97: 1422, 97:
1596, 97: 1752, 97: 1800, 97: 1846, 97: 1932, 97: 1972, 97: 2010, 97:
2080, 97: 2112, 97: 2170, 97: 2196, 97: 2220, 97: 2242, 97: 2262, 97:
2280, 97: 2296, 97: 2310, 97: 2322, 97: 2332, 97: 2340, 97: 2346, 97:
2350, and 97: 2352
From Foo's last statement, we know there must be a unique sum. It is 17, so X and Y must be 13 and 4.
For a discussion of this and similar problems, see http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/logic_sum_product
0, 1, 2, 720!, ?
At first glance, this sequence seems quite wild; 720! is a large number.
But it breaks down quite nicely when you look at the first few
factorials:
0!=1
1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
The key insight that 6!=720. From there it is a matter of simple
substitution:
0, 1, 2, (6!)!,
0, 1, 2, ((3!)!)!,
which would suggest the solution is:
0!!!, 1!!!, 2!!!, 3!!!, 4!!!,...
But 0!!!=1, pointing the way to the actual solution:
0, 1!, 2!!, 3!!!, 4!!!!, ...
Em-Heart-Eight
What comes next? (As seen on the Simpson's. Lisa couldn't solve it,
but Homer could! That's because it is as easy as 1, 2, 3.)
reductio ad absurdum
Try writing numbers using each of the ten digits exactly once so
that the sum of the numbers is exactly 100.
By relaxing
one condition or the other,
19+28+30+7+6+5+4=99
19+28+31+7+6+5+4=100
but after a lot of trial an error, it is
perhaps worth trying to prove that there is no solution:
If we
use two digit numbers, then we'll have whole numbers N (tens) and M
(units) where N*10+M=100 and N+M=45. It follows that N*9=55; Therefore
N=55/9, which is not a whole number. (From Polya)
3 light bulbs
There are three light switches (all in the off position) in the room you are in and three light bulbs in an adjacent room (which cannot be seen from the room you start in). How many trips between the rooms do you need to take in order to map the switches to the light bulbs they control? (A round trip between rooms counts as two trips.)
Two Corks
An example of where the working backwards heuristic helps:
Pencil Flip
7-11
A customer at a 7-11 store bought four items. When the clerk rang up the bill, it came to $7.11. The customer quipped, "Does the bill always come to $7.11 at the 7-11?" The clerk replied, "Of course not. I multiplied the prices of your items and got $7.11." The customer was surprised, "You multiplied the prices? Aren't you supposed to add them?" The clerk apologized, "My mistake. You ate right. I'll add them up. That'll be $7.11 please." What were the prices of the four items?
3 cities
You've got a contract to build roads connecting three cities situated arbitrarily on a plane. How do you minimize the length of the roads you must build?