Answers to last week’s puzzle

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

Class discussion

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!!!!, ...


Heuristic: Look at anomalies for places to attack the problem.

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.)


Heuristic: Don't be distracted from looking for simple solutions first.

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)


Heuristic: If an answer doesn't present itself, think about proving the problem cannot be solved.

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.)


Hueristic: Consider all of the places from whence you can gain a toe-hold on the problem.

Two Corks

An example of where the working backwards heuristic helps:

move the corks from here

to here


Heuristic: Try working the problem backwards.

Pencil Flip

Without removing either hand from the pencil, flip your thumbs below the pencil and your fingers above it.

Puzzles for next week

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?