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A brute force approach to find multipliers for the first part of Peter Winkler's math puzzle 'Zeros, ones or twos', in his book _Mathematical Puzzles_
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# A horrible awful brute-force approach to Peter Winkler's 'Zeroes, ones and twos' problem, part 1 | |
(1..100).each do |p| | |
found = false | |
q = 0 | |
until found | |
q += 1 | |
product = p * q | |
if !(product.to_s =~ /[^01]/) || q > 1000000000 | |
found = true | |
p "#{p} * #{q} = #{product}" | |
end | |
end | |
end |
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# A slightly less brute-force approach to Peter Winkler's 'Zeroes, ones and twos' problem, part 1 | |
# this time it only checks each multiplier for the remaining undiscovered numbers | |
require 'pp' | |
pairs = {} | |
my_range = (1..1000) | |
partners = my_range.to_a | |
q = 0 | |
until partners.empty? || q > 100000000000 | |
q += 1 | |
hits , misses = partners.partition {|p| !((p * q).to_s =~ /[^01]/)} | |
hits.each do |p| | |
pairs[p]=[q,p*q] | |
p "#{pairs.length}: #{p} * #{q} = #{p*q}" | |
end | |
partners = misses | |
end | |
pp pairs.sort |
And the smallest k
, such that n*k
contains only even digits:
[[1, [2, 2]],
[2, [1, 2]],
[3, [2, 6]],
[4, [1, 4]],
[5, [4, 20]],
[6, [1, 6]],
[7, [4, 28]],
[8, [1, 8]],
[9, [32, 288]],
[10, [2, 20]],
[11, [2, 22]],
[12, [2, 24]],
[13, [2, 26]],
[14, [2, 28]],
[15, [4, 60]],
[16, [3, 48]],
[17, [4, 68]],
[18, [16, 288]],
[19, [12, 228]],
[20, [1, 20]],
[21, [2, 42]],
[22, [1, 22]],
[23, [2, 46]],
[24, [1, 24]],
[25, [8, 200]],
[26, [1, 26]],
[27, [18, 486]],
[28, [1, 28]],
[29, [14, 406]],
[30, [2, 60]],
[31, [2, 62]],
[32, [2, 64]],
[33, [2, 66]],
[34, [2, 68]],
[35, [8, 280]],
[36, [8, 288]],
[37, [6, 222]],
[38, [6, 228]],
[39, [12, 468]],
[40, [1, 40]],
[41, [2, 82]],
[42, [1, 42]],
[43, [2, 86]],
[44, [1, 44]],
[45, [64, 2880]],
[46, [1, 46]],
[47, [6, 282]],
[48, [1, 48]],
[49, [14, 686]],
[50, [4, 200]],
[51, [4, 204]],
[52, [4, 208]],
[53, [8, 424]],
[54, [9, 486]],
[55, [4, 220]],
[56, [4, 224]],
[57, [4, 228]],
[58, [7, 406]],
[59, [14, 826]],
[60, [1, 60]],
[61, [4, 244]],
[62, [1, 62]],
[63, [14, 882]],
[64, [1, 64]],
[65, [4, 260]],
[66, [1, 66]],
[67, [4, 268]],
[68, [1, 68]],
[69, [12, 828]],
[70, [4, 280]],
[71, [4, 284]],
[72, [4, 288]],
[73, [28, 2044]],
[74, [3, 222]],
[75, [8, 600]],
[76, [3, 228]],
[77, [6, 462]],
[78, [6, 468]],
[79, [34, 2686]],
[80, [1, 80]],
[81, [6, 486]],
[82, [1, 82]],
[83, [8, 664]],
[84, [1, 84]],
[85, [8, 680]],
[86, [1, 86]],
[87, [24, 2088]],
[88, [1, 88]],
[89, [32, 2848]],
[90, [32, 2880]],
[91, [22, 2002]],
[92, [5, 460]],
[93, [22, 2046]],
[94, [3, 282]],
[95, [24, 2280]],
[96, [3, 288]],
[97, [44, 4268]],
[98, [7, 686]],
[99, [2312, 228888]],
[100, [2, 200]],
[101, [2, 202]],
[102, [2, 204]],
[103, [2, 206]],
[104, [2, 208]],
[105, [4, 420]],
[106, [4, 424]],
[107, [4, 428]],
[108, [6, 648]],
[109, [74, 8066]],
[110, [2, 220]],
[111, [2, 222]],
[112, [2, 224]],
[113, [2, 226]],
[114, [2, 228]],
[115, [4, 460]],
[116, [4, 464]],
[117, [4, 468]],
[118, [7, 826]],
[119, [34, 4046]],
[120, [2, 240]],
[121, [2, 242]],
[122, [2, 244]],
[123, [2, 246]],
[124, [2, 248]],
[125, [16, 2000]],
[126, [7, 882]],
[127, [18, 2286]],
[128, [5, 640]],
[129, [16, 2064]],
[130, [2, 260]],
[131, [2, 262]],
[132, [2, 264]],
[133, [2, 266]],
[134, [2, 268]],
[135, [36, 4860]],
[136, [3, 408]],
[137, [6, 822]],
[138, [6, 828]],
[139, [16, 2224]],
[140, [2, 280]],
[141, [2, 282]],
[142, [2, 284]],
[143, [2, 286]],
[144, [2, 288]],
[145, [28, 4060]],
[146, [14, 2044]],
[147, [6, 882]],
[148, [3, 444]],
[149, [14, 2086]],
[150, [4, 600]],
[151, [4, 604]],
[152, [4, 608]],
[153, [16, 2448]],
[154, [3, 462]],
[155, [4, 620]],
[156, [3, 468]],
[157, [4, 628]],
[158, [17, 2686]],
[159, [14, 2226]],
[160, [3, 480]],
[161, [4, 644]],
[162, [3, 486]],
[163, [14, 2282]],
[164, [5, 820]],
[165, [4, 660]],
[166, [4, 664]],
[167, [4, 668]],
[168, [5, 840]],
[169, [12, 2028]],
[170, [4, 680]],
[171, [4, 684]],
[172, [4, 688]],
[173, [14, 2422]],
[174, [12, 2088]],
[175, [16, 2800]],
[176, [5, 880]],
[177, [24, 4248]],
[178, [16, 2848]],
[179, [16, 2864]],
[180, [16, 2880]],
[181, [48, 8688]],
[182, [11, 2002]],
[183, [22, 4026]],
[184, [11, 2024]],
[185, [12, 2220]],
[186, [11, 2046]],
[187, [12, 2244]],
[188, [11, 2068]],
[189, [12, 2268]],
[190, [12, 2280]],
[191, [22, 4202]],
[192, [14, 2688]],
[193, [22, 4246]],
[194, [22, 4268]],
[195, [24, 4680]],
[196, [33, 6468]],
[197, [44, 8668]],
[198, [1156, 228888]],
[199, [112, 22288]],
[200, [1, 200]],
[201, [2, 402]],
[202, [1, 202]],
[203, [2, 406]],
[204, [1, 204]],
[205, [4, 820]],
[206, [1, 206]],
[207, [4, 828]],
[208, [1, 208]],
[209, [32, 6688]],
[210, [2, 420]],
[211, [2, 422]],
[212, [2, 424]],
[213, [2, 426]],
[214, [2, 428]],
[215, [4, 860]],
[216, [3, 648]],
[217, [4, 868]],
[218, [37, 8066]],
[219, [12, 2628]],
[220, [1, 220]],
[221, [2, 442]],
[222, [1, 222]],
[223, [2, 446]],
[224, [1, 224]],
[225, [128, 28800]],
[226, [1, 226]],
[227, [18, 4086]],
[228, [1, 228]],
[229, [36, 8244]],
[230, [2, 460]],
[231, [2, 462]],
[232, [2, 464]],
[233, [2, 466]],
[234, [2, 468]],
[235, [12, 2820]],
[236, [18, 4248]],
[237, [12, 2844]],
[238, [17, 4046]],
[239, [12, 2868]],
[240, [1, 240]],
[241, [2, 482]],
[242, [1, 242]],
[243, [2, 486]],
[244, [1, 244]],
[245, [28, 6860]],
[246, [1, 246]],
[247, [18, 4446]],
[248, [1, 248]],
[249, [18, 4482]],
[250, [8, 2000]],
[251, [8, 2008]],
[252, [9, 2268]],
[253, [8, 2024]],
[254, [9, 2286]],
[255, [8, 2040]],
[256, [8, 2048]],
[257, [18, 4626]],
[258, [8, 2064]],
[259, [18, 4662]],
[260, [1, 260]],
[261, [8, 2088]],
[262, [1, 262]],
[263, [16, 4208]],
[264, [1, 264]],
[265, [16, 4240]],
[266, [1, 266]],
[267, [18, 4806]],
[268, [1, 268]],
[269, [18, 4842]],
[270, [18, 4860]],
[271, [82, 22222]],
[272, [9, 2448]],
[273, [22, 6006]],
[274, [3, 822]],
[275, [8, 2200]],
[276, [3, 828]],
[277, [24, 6648]],
[278, [8, 2224]],
[279, [16, 4464]],
[280, [1, 280]],
[281, [8, 2248]],
[282, [1, 282]],
[283, [8, 2264]],
[284, [1, 284]],
[285, [8, 2280]],
[286, [1, 286]],
[287, [24, 6888]],
[288, [1, 288]],
[289, [14, 4046]],
[290, [14, 4060]],
[291, [22, 6402]],
[292, [7, 2044]],
[293, [16, 4688]],
[294, [3, 882]],
[295, [28, 8260]],
[296, [3, 888]],
[297, [838, 248886]],
[298, [7, 2086]],
[299, [134, 40066]],
[300, [2, 600]],
[301, [2, 602]],
[302, [2, 604]],
[303, [2, 606]],
[304, [2, 608]],
[305, [8, 2440]],
[306, [8, 2448]],
[307, [66, 20262]],
[308, [8, 2464]],
[309, [72, 22248]],
[310, [2, 620]],
[311, [2, 622]],
[312, [2, 624]],
[313, [2, 626]],
[314, [2, 628]],
[315, [28, 8820]],
[316, [9, 2844]],
[317, [26, 8242]],
[318, [7, 2226]],
[319, [14, 4466]],
[320, [2, 640]],
[321, [2, 642]],
[322, [2, 644]],
[323, [2, 646]],
[324, [2, 648]],
[325, [8, 2600]],
[326, [7, 2282]],
[327, [186, 60822]],
[328, [8, 2624]],
[329, [14, 4606]],
[330, [2, 660]],
[331, [2, 662]],
[332, [2, 664]],
[333, [2, 666]],
[334, [2, 668]],
[335, [8, 2680]],
[336, [8, 2688]],
[337, [6, 2022]],
[338, [6, 2028]],
[339, [12, 4068]],
[340, [2, 680]],
[341, [2, 682]],
[342, [2, 684]],
[343, [2, 686]],
[344, [2, 688]],
[345, [24, 8280]],
[346, [7, 2422]],
[347, [6, 2082]],
[348, [6, 2088]],
[349, [14, 4886]],
[350, [8, 2800]],
[351, [8, 2808]],
[352, [7, 2464]],
[353, [8, 2824]],
[354, [12, 4248]],
[355, [8, 2840]],
[356, [8, 2848]],
[357, [12, 4284]],
[358, [8, 2864]],
[359, [18, 6462]],
[360, [8, 2880]],
[361, [8, 2888]],
[362, [24, 8688]],
[363, [68, 24684]],
[364, [11, 4004]],
[365, [56, 20440]],
[366, [11, 4026]],
[367, [6, 2202]],
[368, [6, 2208]],
[369, [12, 4428]],
[370, [6, 2220]],
[371, [6, 2226]],
[372, [7, 2604]],
[373, [22, 8206]],
[374, [6, 2244]],
[375, [16, 6000]],
[376, [13, 4888]],
[377, [6, 2262]],
[378, [6, 2268]],
[379, [16, 6064]],
[380, [6, 2280]],
[381, [6, 2286]],
[382, [11, 4202]],
[383, [22, 8426]],
[384, [7, 2688]],
[385, [12, 4620]],
[386, [11, 4246]],
[387, [12, 4644]],
[388, [11, 4268]],
[389, [12, 4668]],
[390, [12, 4680]],
[391, [22, 8602]],
[392, [17, 6664]],
[393, [16, 6288]],
[394, [22, 8668]],
[395, [68, 26860]],
[396, [578, 228888]],
[397, [52, 20644]],
[398, [56, 22288]],
[399, [112, 44688]],
[400, [1, 400]],
[401, [2, 802]],
[402, [1, 402]],
[403, [2, 806]],
[404, [1, 404]],
[405, [12, 4860]],
[406, [1, 406]],
[407, [6, 2442]],
[408, [1, 408]],
[409, [54, 22086]],
[410, [2, 820]],
[411, [2, 822]],
[412, [2, 824]],
[413, [2, 826]],
[414, [2, 828]],
[415, [16, 6640]],
[416, [5, 2080]],
[417, [64, 26688]],
[418, [16, 6688]],
[419, [54, 22626]],
[420, [1, 420]],
[421, [2, 842]],
[422, [1, 422]],
[423, [2, 846]],
[424, [1, 424]],
[425, [16, 6800]],
[426, [1, 426]],
[427, [52, 22204]],
[428, [1, 428]],
[429, [14, 6006]],
[430, [2, 860]],
[431, [2, 862]],
[432, [2, 864]],
[433, [2, 866]],
[434, [2, 868]],
[435, [48, 20880]],
[436, [19, 8284]],
[437, [6, 2622]],
[438, [6, 2628]],
[439, [52, 22828]],
[440, [1, 440]],
[441, [2, 882]],
[442, [1, 442]],
[443, [2, 886]],
[444, [1, 444]],
[445, [64, 28480]],
[446, [1, 446]],
[447, [6, 2682]],
[448, [1, 448]],
[449, [14, 6286]],
[450, [64, 28800]],
[451, [64, 28864]],
[452, [5, 2260]],
[453, [54, 24462]],
[454, [9, 4086]],
[455, [44, 20020]],
[456, [5, 2280]],
[457, [18, 8226]],
[458, [18, 8244]],
[459, [14, 6426]],
[460, [1, 460]],
[461, [44, 20284]],
[462, [1, 462]],
[463, [14, 6482]],
[464, [1, 464]],
[465, [44, 20460]],
[466, [1, 466]],
[467, [6, 2802]],
[468, [1, 468]],
[469, [18, 8442]],
[470, [6, 2820]],
[471, [6, 2826]],
[472, [9, 4248]],
[473, [14, 6622]],
[474, [6, 2844]],
[475, [48, 22800]],
[476, [9, 4284]],
[477, [6, 2862]],
[478, [6, 2868]],
[479, [18, 8622]],
[480, [1, 480]],
[481, [6, 2886]],
[482, [1, 482]],
[483, [42, 20286]],
[484, [1, 484]],
[485, [88, 42680]],
[486, [1, 486]],
[487, [46, 22402]],
[488, [1, 488]],
[489, [14, 6846]],
[490, [14, 6860]],
[491, [42, 20622]],
[492, [5, 2460]],
[493, [54, 26622]],
[494, [9, 4446]],
[495, [4624, 2288880]],
[496, [5, 2480]],
[497, [46, 22862]],
[498, [9, 4482]],
[499, [134, 66866]],
[500, [4, 2000]],
[501, [4, 2004]],
[502, [4, 2008]],
[503, [8, 4024]],
[504, [12, 6048]],
[505, [4, 2020]],
[506, [4, 2024]],
[507, [4, 2028]],
[508, [8, 4064]],
[509, [52, 26468]],
[510, [4, 2040]],
[511, [4, 2044]],
[512, [4, 2048]],
[513, [16, 8208]],
[514, [9, 4626]],
[515, [4, 2060]],
[516, [4, 2064]],
[517, [4, 2068]],
[518, [9, 4662]],
[519, [12, 6228]],
[520, [4, 2080]],
[521, [4, 2084]],
[522, [4, 2088]],
[523, [54, 28242]],
[524, [5, 2620]],
[525, [8, 4200]],
[526, [8, 4208]],
[527, [38, 20026]],
[528, [5, 2640]],
[529, [16, 8464]],
[530, [8, 4240]],
[531, [8, 4248]],
[532, [5, 2660]],
[533, [8, 4264]],
[534, [9, 4806]],
[535, [8, 4280]],
[536, [5, 2680]],
[537, [12, 6444]],
[538, [9, 4842]],
[539, [12, 6468]],
[540, [9, 4860]],
[541, [46, 24886]],
[542, [41, 22222]],
[543, [16, 8688]],
[544, [42, 22848]],
[545, [148, 80660]],
[546, [11, 6006]],
[547, [44, 24068]],
[548, [11, 6028]],
[549, [38, 20862]],
[550, [4, 2200]],
[551, [4, 2204]],
[552, [4, 2208]],
[553, [8, 4424]],
[554, [12, 6648]],
[555, [4, 2220]],
[556, [4, 2224]],
[557, [4, 2228]],
[558, [8, 4464]],
[559, [72, 40248]],
[560, [4, 2240]],
[561, [4, 2244]],
[562, [4, 2248]],
[563, [36, 20268]],
[564, [5, 2820]],
[565, [4, 2260]],
[566, [4, 2264]],
[567, [4, 2268]],
[568, [5, 2840]],
[569, [12, 6828]],
[570, [4, 2280]],
[571, [4, 2284]],
[572, [4, 2288]],
[573, [14, 8022]],
[574, [12, 6888]],
[575, [8, 4600]],
[576, [5, 2880]],
[577, [84, 48468]],
[578, [7, 4046]],
[579, [36, 20844]],
[580, [7, 4060]],
[581, [8, 4648]],
[582, [11, 6402]],
[583, [8, 4664]],
[584, [7, 4088]],
[585, [8, 4680]],
[586, [8, 4688]],
[587, [72, 42264]],
[588, [11, 6468]],
[589, [14, 8246]],
[590, [14, 8260]],
[591, [42, 24822]],
[592, [14, 8288]],
[593, [48, 28464]],
[594, [419, 248886]],
[595, [68, 40460]],
[596, [34, 20264]],
[597, [38, 22686]],
[598, [67, 40066]],
[599, [114, 68286]],
[600, [1, 600]],
[601, [4, 2404]],
[602, [1, 602]],
[603, [8, 4824]],
[604, [1, 604]],
[605, [4, 2420]],
[606, [1, 606]],
[607, [4, 2428]],
[608, [1, 608]],
[609, [76, 46284]],
[610, [4, 2440]],
[611, [4, 2444]],
[612, [4, 2448]],
[613, [34, 20842]],
[614, [33, 20262]],
[615, [4, 2460]],
[616, [4, 2464]],
[617, [4, 2468]],
[618, [36, 22248]],
[619, [14, 8666]],
[620, [1, 620]],
[621, [4, 2484]],
[622, [1, 622]],
[623, [36, 22428]],
[624, [1, 624]],
[625, [32, 20000]],
[626, [1, 626]],
[627, [32, 20064]],
[628, [1, 628]],
[629, [14, 8806]],
[630, [14, 8820]],
[631, [106, 66886]],
[632, [7, 4424]],
[633, [14, 8862]],
[634, [13, 8242]],
[635, [36, 22860]],
[636, [13, 8268]],
[637, [38, 24206]],
[638, [7, 4466]],
[639, [32, 20448]],
[640, [1, 640]],
[641, [44, 28204]],
[642, [1, 642]],
[643, [76, 48868]],
[644, [1, 644]],
[645, [32, 20640]],
[646, [1, 646]],
[647, [44, 28468]],
[648, [1, 648]],
[649, [34, 22066]],
[650, [4, 2600]],
[651, [4, 2604]],
[652, [4, 2608]],
[653, [34, 22202]],
[654, [93, 60822]],
[655, [4, 2620]],
[656, [4, 2624]],
[657, [4, 2628]],
[658, [7, 4606]],
[659, [34, 22406]],
[660, [1, 660]],
[661, [4, 2644]],
[662, [1, 662]],
[663, [122, 80886]],
[664, [1, 664]],
[665, [4, 2660]],
[666, [1, 666]],
[667, [4, 2668]],
[668, [1, 668]],
[669, [12, 8028]],
[670, [4, 2680]],
[671, [4, 2684]],
[672, [4, 2688]],
[673, [34, 22882]],
[674, [3, 2022]],
[675, [72, 48600]],
[676, [3, 2028]],
[677, [6, 4062]],
[678, [6, 4068]],
[679, [36, 24444]],
[680, [1, 680]],
[681, [6, 4086]],
[682, [1, 682]],
[683, [42, 28686]],
[684, [1, 684]],
[685, [12, 8220]],
[686, [1, 686]],
[687, [12, 8244]],
[688, [1, 688]],
[689, [12, 8268]],
[690, [12, 8280]],
[691, [62, 42842]],
[692, [7, 4844]],
[693, [388, 268884]],
[694, [3, 2082]],
[695, [32, 22240]],
[696, [3, 2088]],
[697, [38, 26486]],
[698, [7, 4886]],
[699, [118, 82482]],
[700, [4, 2800]],
[701, [4, 2804]],
[702, [4, 2808]],
[703, [94, 66082]],
[704, [6, 4224]],
[705, [4, 2820]],
[706, [4, 2824]],
[707, [4, 2828]],
[708, [6, 4248]],
[709, [32, 22688]],
[710, [4, 2840]],
[711, [4, 2844]],
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And while we're at it, here are the numbers such that the product has no digits larger than
4
: