by Guy Jacobson and Matt Gruskin

The flavor text of the puzzle is:

In Patriots’ Day Town, everyone roots for New England and uses graphs to analyze their play.

Solvers need to resist any urge they might have to interpret graphs in the sense of an x–y plot of the polynomials—that’s not what they are meant to do.

First, solvers should find all the roots of the polynomials (via programming, or any number of symbolic algebra packages, or by hand if they are masochists). They will discover something interesting: each polynomial has three distinct integral roots, and every root is shared by exactly two polynomials. Most of the roots are positive integers, but there are also some negatives.

The negative roots are the integers from [-38, -1] with no gaps. The positive roots are also mostly consecutive, but there are some gaps of length one. Counting the lengths of the runs of consecutive roots (between gaps) in order yields the following sequence:

14, 1, 13, 5, 19, 6, 15, 18, 20, 8, 5, 7, 18, 1, 16, 8, 19

Interpreting these numbers as alphabetic indices spells the message:

NAMES FOR THE GRAPHS

which is an instruction telling solvers to find out what the graphs are named. But what graphs are they supposed to look at?

Solvers need to think about combinatorial graphs rather than x–y plots. Each polynomial corresponds to a vertex in a graph, with an edge between two vertices if they share a root. Since each graph has exactly three roots, the resulting graph will be a 3-regular, or cubic graph. If solvers look at the connectivity pattern among the vertices, they will observe that the full configuration separates by connected components into ten different cubic graphs.

All ten graphs are instances of well-known named graphs. Solvers may need to work out some simple graph isomorphism problems to figure out the which graph from the polynomials is which named graph. This can be accomplished by looking at the number of nodes/edges, visualizing the graphs laid out on the plane and staring at the output, writing code, etc.

Here are the ten graphs and their names; edges are labeled with the values of the shared roots. (Apologies that some of the edge labels are obscured and that some layouts are a bit wonky):

## BIDIAKIS CUBE ## COXETER ## FRANKLIN ## FRUCHT ## HEAWOOD ## MARKSTROM ## MOBIUS-KANTOR ## PAPPUS ## PETERSEN ## WAGNER Now what? Solvers need to observe that while the set of positive roots associated with each graph look pretty random, the negative roots are always consecutive integers. Solvers can tally the negative roots used in each graph and order by the range of negative integers. Then solvers can use the count of negative roots as an index into the name of the corresponding graph to extract a letter:

Order Graph Name Negative Roots # of Negative Roots Letter
1 COXETER -1 to -3 3 X
2 PETERSEN -4 to -7 4 E
3 WAGNER -8 to -11 4 N
4 MOBIUS-KANTOR -12 to -13 2 O
5 PAPPUS -14 to -17 4 P
6 HEAWOOD -18 1 H
7 MARKSTROM -19 to -26 8 O
8 BIDIAKIS CUBE -27 1 B
9 FRANKLIN -28 to -34 7 I
10 FRUCHT -35 to -38 4 C

This yields the puzzle answer, XENOPHOBIC.

The following table identifies the roots of all of the polynomials, and the graphs to which they belong:

Graph Name Roots Polynomial
BIDIAKIS CUBE -27 13 30 $$x^3-16x^2-771x+10530$$
BIDIAKIS CUBE -27 34 197 $$x^3-204x^2+461x+180846$$
BIDIAKIS CUBE 12 14 123 $$x^3-149x^2+3366x-20664$$
BIDIAKIS CUBE 12 52 157 $$x^3-221x^2+10672x-97968$$
BIDIAKIS CUBE 13 94 177 $$x^3-284x^2+20161x-216294$$
BIDIAKIS CUBE 14 58 197 $$x^3-269x^2+14996x-159964$$
BIDIAKIS CUBE 30 45 76 $$x^3-151x^2+7050x-102600$$
BIDIAKIS CUBE 34 52 76 $$x^3-162x^2+8304x-134368$$
BIDIAKIS CUBE 45 134 157 $$x^3-336x^2+34133x-946710$$
BIDIAKIS CUBE 58 156 177 $$x^3-391x^2+46926x-1601496$$
BIDIAKIS CUBE 94 113 134 $$x^3-341x^2+38360x-1423348$$
BIDIAKIS CUBE 113 123 156 $$x^3-392x^2+50715x-2168244$$
COXETER -3 23 93 $$x^3-113x^2+1791x+6417$$
COXETER -3 56 110 $$x^3-163x^2+5662x+18480$$
COXETER -2 73 171 $$x^3-242x^2+11995x+24966$$
COXETER -2 119 189 $$x^3-306x^2+21875x+44982$$
COXETER -1 4 73 $$x^3-76x^2+215x+292$$
COXETER -1 150 206 $$x^3-355x^2+30544x+30900$$
COXETER 2 27 125 $$x^3-154x^2+3679x-6750$$
COXETER 2 38 63 $$x^3-103x^2+2596x-4788$$
COXETER 4 166 173 $$x^3-343x^2+30074x-114872$$
COXETER 18 24 137 $$x^3-179x^2+6186x-59184$$
COXETER 18 67 166 $$x^3-251x^2+15316x-200196$$
COXETER 23 102 194 $$x^3-319x^2+26596x-455124$$
COXETER 24 27 126 $$x^3-177x^2+7074x-81648$$
COXETER 38 54 206 $$x^3-298x^2+21004x-422712$$
COXETER 39 67 124 $$x^3-230x^2+15757x-324012$$
COXETER 39 110 133 $$x^3-282x^2+24107x-570570$$
COXETER 54 72 191 $$x^3-317x^2+27954x-742608$$
COXETER 56 63 145 $$x^3-264x^2+20783x-511560$$
COXETER 72 119 133 $$x^3-324x^2+33971x-1139544$$
COXETER 78 102 117 $$x^3-297x^2+29016x-930852$$
COXETER 78 137 191 $$x^3-406x^2+51751x-2041026$$
COXETER 93 126 171 $$x^3-390x^2+49167x-2003778$$
COXETER 106 117 170 $$x^3-393x^2+50312x-2108340$$
COXETER 106 145 173 $$x^3-424x^2+58793x-2659010$$
COXETER 124 144 159 $$x^3-427x^2+60468x-2839104$$
COXETER 125 141 144 $$x^3-410x^2+55929x-2538000$$
COXETER 141 170 189 $$x^3-500x^2+82749x-4530330$$
COXETER 150 159 194 $$x^3-503x^2+83796x-4626900$$
FRANKLIN -34 -29 9 $$x^3+54x^2+419x-8874$$
FRANKLIN -34 -28 21 $$x^3+41x^2-350x-19992$$
FRANKLIN -33 -32 -28 $$x^3+93x^2+2876x+29568$$
FRANKLIN -33 -30 90 $$x^3-27x^2-4680x-89100$$
FRANKLIN -32 116 185 $$x^3-269x^2+11828x+686720$$
FRANKLIN -31 9 127 $$x^3-105x^2-3073x+35433$$
FRANKLIN -31 116 186 $$x^3-271x^2+12214x+668856$$
FRANKLIN -30 104 185 $$x^3-259x^2+10570x+577200$$
FRANKLIN -29 68 104 $$x^3-143x^2+2084x+205088$$
FRANKLIN 21 68 183 $$x^3-272x^2+17715x-261324$$
FRANKLIN 90 127 172 $$x^3-389x^2+48754x-1965960$$
FRANKLIN 172 183 186 $$x^3-541x^2+97506x-5854536$$
FRUCHT -38 70 202 $$x^3-234x^2+3804x+537320$$
FRUCHT -38 168 178 $$x^3-308x^2+16756x+1136352$$
FRUCHT -37 46 69 $$x^3-78x^2-1081x+117438$$
FRUCHT -37 70 168 $$x^3-201x^2+2954x+435120$$
FRUCHT -36 -35 139 $$x^3-68x^2-8609x-175140$$
FRUCHT -36 82 182 $$x^3-228x^2+5420x+537264$$
FRUCHT -35 202 208 $$x^3-375x^2+27666x+1470560$$
FRUCHT 44 69 178 $$x^3-291x^2+23150x-540408$$
FRUCHT 44 89 182 $$x^3-315x^2+28122x-712712$$
FRUCHT 46 114 179 $$x^3-339x^2+33884x-938676$$
FRUCHT 82 89 114 $$x^3-285x^2+26792x-831972$$
FRUCHT 139 179 208 $$x^3-526x^2+91025x-5175248$$
HEAWOOD -18 71 96 $$x^3-149x^2+3810x+122688$$
HEAWOOD -18 103 207 $$x^3-292x^2+15741x+383778$$
HEAWOOD 11 100 101 $$x^3-212x^2+12311x-111100$$
HEAWOOD 11 105 149 $$x^3-265x^2+18439x-172095$$
HEAWOOD 25 61 196 $$x^3-282x^2+18381x-298900$$
HEAWOOD 25 66 131 $$x^3-222x^2+13571x-216150$$
HEAWOOD 26 101 199 $$x^3-326x^2+27899x-522574$$
HEAWOOD 26 192 196 $$x^3-414x^2+47720x-978432$$
HEAWOOD 55 96 199 $$x^3-350x^2+35329x-1050720$$
HEAWOOD 55 131 175 $$x^3-361x^2+39755x-1260875$$
HEAWOOD 61 71 149 $$x^3-281x^2+23999x-645319$$
HEAWOOD 66 100 103 $$x^3-269x^2+23698x-679800$$
HEAWOOD 97 105 175 $$x^3-377x^2+45535x-1782375$$
HEAWOOD 97 192 207 $$x^3-496x^2+78447x-3855168$$
MARKSTROM -26 -20 121 $$x^3-75x^2-5046x-62920$$
MARKSTROM -26 29 109 $$x^3-112x^2-427x+82186$$
MARKSTROM -25 -24 6 $$x^3+43x^2+306x-3600$$
MARKSTROM -25 -19 20 $$x^3+24x^2-405x-9500$$
MARKSTROM -24 20 95 $$x^3-91x^2-860x+45600$$
MARKSTROM -23 6 158 $$x^3-141x^2-2824x+21804$$
MARKSTROM -23 111 205 $$x^3-293x^2+15487x+523365$$
MARKSTROM -22 53 79 $$x^3-110x^2+1283x+92114$$
MARKSTROM -22 88 140 $$x^3-206x^2+7304x+271040$$
MARKSTROM -21 47 59 $$x^3-85x^2+547x+58233$$
MARKSTROM -21 51 111 $$x^3-141x^2+2259x+118881$$
MARKSTROM -20 86 88 $$x^3-154x^2+4088x+151360$$
MARKSTROM -19 1 130 $$x^3-112x^2-2359x+2470$$
MARKSTROM 1 19 148 $$x^3-168x^2+2979x-2812$$
MARKSTROM 10 53 148 $$x^3-211x^2+9854x-78440$$
MARKSTROM 10 60 83 $$x^3-153x^2+6410x-49800$$
MARKSTROM 19 130 158 $$x^3-307x^2+26012x-390260$$
MARKSTROM 29 187 188 $$x^3-404x^2+46031x-1019524$$
MARKSTROM 47 60 155 $$x^3-262x^2+19405x-437100$$
MARKSTROM 51 188 205 $$x^3-444x^2+58583x-1965540$$
MARKSTROM 59 109 187 $$x^3-355x^2+37847x-1202597$$
MARKSTROM 79 140 195 $$x^3-414x^2+53765x-2156700$$
MARKSTROM 83 155 195 $$x^3-433x^2+59275x-2508675$$
MARKSTROM 86 95 121 $$x^3-302x^2+30071x-988570$$
MOBIUS-KANTOR -13 -12 8 $$x^3+17x^2-44x-1248$$
MOBIUS-KANTOR -13 43 87 $$x^3-117x^2+2051x+48633$$
MOBIUS-KANTOR -12 108 147 $$x^3-243x^2+12816x+190512$$
MOBIUS-KANTOR 7 16 152 $$x^3-175x^2+3608x-17024$$
MOBIUS-KANTOR 7 85 198 $$x^3-290x^2+18811x-117810$$
MOBIUS-KANTOR 8 16 161 $$x^3-185x^2+3992x-20608$$
MOBIUS-KANTOR 35 75 163 $$x^3-273x^2+20555x-427875$$
MOBIUS-KANTOR 35 77 92 $$x^3-204x^2+12999x-247940$$
MOBIUS-KANTOR 43 85 98 $$x^3-226x^2+16199x-358190$$
MOBIUS-KANTOR 48 49 169 $$x^3-266x^2+18745x-397488$$
MOBIUS-KANTOR 48 87 138 $$x^3-273x^2+22806x-576288$$
MOBIUS-KANTOR 49 147 200 $$x^3-396x^2+46403x-1440600$$
MOBIUS-KANTOR 75 108 198 $$x^3-381x^2+44334x-1603800$$
MOBIUS-KANTOR 77 98 200 $$x^3-375x^2+42546x-1509200$$
MOBIUS-KANTOR 92 138 152 $$x^3-382x^2+47656x-1929792$$
MOBIUS-KANTOR 161 163 169 $$x^3-493x^2+80999x-4435067$$
PAPPUS -17 42 62 $$x^3-87x^2+836x+44268$$
PAPPUS -17 65 184 $$x^3-232x^2+7727x+203320$$
PAPPUS -16 -15 36 $$x^3-5x^2-876x-8640$$
PAPPUS -16 42 180 $$x^3-206x^2+4008x+120960$$
PAPPUS -15 165 193 $$x^3-343x^2+26475x+477675$$
PAPPUS -14 36 122 $$x^3-144x^2+2180x+61488$$
PAPPUS -14 50 118 $$x^3-154x^2+3548x+82600$$
PAPPUS 5 128 146 $$x^3-279x^2+20058x-93440$$
PAPPUS 5 165 174 $$x^3-344x^2+30405x-143550$$
PAPPUS 33 74 180 $$x^3-287x^2+21702x-439560$$
PAPPUS 33 174 209 $$x^3-416x^2+49005x-1200078$$
PAPPUS 41 50 132 $$x^3-223x^2+14062x-270600$$
PAPPUS 41 65 209 $$x^3-315x^2+24819x-556985$$
PAPPUS 62 112 154 $$x^3-328x^2+33740x-1069376$$
PAPPUS 74 118 136 $$x^3-328x^2+34844x-1187552$$
PAPPUS 112 136 146 $$x^3-394x^2+51440x-2223872$$
PAPPUS 122 128 184 $$x^3-434x^2+61616x-2873344$$
PAPPUS 132 154 193 $$x^3-479x^2+75526x-3923304$$
PETERSEN -7 3 115 $$x^3-111x^2-481x+2415$$
PETERSEN -7 91 153 $$x^3-237x^2+12215x+97461$$
PETERSEN -6 22 84 $$x^3-100x^2+1212x+11088$$
PETERSEN -6 142 151 $$x^3-287x^2+19684x+128652$$
PETERSEN -5 22 153 $$x^3-170x^2+2491x+16830$$
PETERSEN -5 107 176 $$x^3-278x^2+17417x+94160$$
PETERSEN -4 91 142 $$x^3-229x^2+11990x+51688$$
PETERSEN -4 176 203 $$x^3-375x^2+34212x+142912$$
PETERSEN 3 84 203 $$x^3-290x^2+17913x-51156$$
PETERSEN 107 115 151 $$x^3-373x^2+45827x-1858055$$
WAGNER -11 -10 -9 $$x^3+30x^2+299x+990$$
WAGNER -11 -8 201 $$x^3-182x^2-3731x-17688$$
WAGNER -10 160 167 $$x^3-317x^2+23450x+267200$$
WAGNER -9 81 204 $$x^3-276x^2+13959x+148716$$
WAGNER -8 28 160 $$x^3-180x^2+2976x+35840$$
WAGNER 28 40 81 $$x^3-149x^2+6628x-90720$$
WAGNER 32 40 201 $$x^3-273x^2+15752x-257280$$
WAGNER 32 167 204 $$x^3-403x^2+45940x-1090176$$