Datasets:
problem_id stringlengths 7 7 | question stringlengths 1.37k 2.42k | image_path stringlengths 23 23 | options stringclasses 0
values | answer stringlengths 1 161 | category stringclasses 1
value | subcategory stringclasses 1
value | explanation stringlengths 429 4.63k | difficulty stringclasses 2
values | hard bool 2
classes | image imagewidth (px) 290 4.1k |
|---|---|---|---|---|---|---|---|---|---|---|
GT-0004 | Fix \(\lambda\) as the partition encoded by the displayed left-justified array of congruent square cells in French convention. Every outlined square is exactly one cell, each shared line segment only separates neighboring cells, and the array is complete, with no cell omitted, hidden, labeled, shaded, or supplied by su... | data/images/GT-0004.png | null | \(\left(3^6\cdot 7^2\cdot 13,\sqrt{55}\right)\) | 数学 | 图论 | Reading the row widths from bottom to top gives the partition
$$\lambda=(6,4,2,2,1,1).$$
Consequently, the total number of cells and the symmetric-group degree are
$$n=|\lambda|=6+4+2+2+1+1=16.$$
Counting the cells in each column gives the conjugate partition
$$\lambda'_j=\#\{i:\lambda_i\ge j\},\qquad \lambda'=(6,4,2,2... | Medium | false | |
GT-0008 | Set $G_D$ to be the finite simple graph constructed from the given shaded planar cell complex by the incidence rules below. The pink region marked $D$ has a boundary consisting of one solid red polygonal chain together with one connected dotted red arc. A black rosette at a point means the local star formed by the blac... | data/images/GT-0008.png | null | $(11,11)$ | 数学 | 图论 | The local branch criterion excludes corners that are only bends or crossings with black sides. Along the prescribed traversal, the qualifying centers are the top center, the upper-right center, the lower-right center, the low center, and the lower-left center. Hence
$$(h_1,h_2,h_3,h_4,h_5),\qquad n=5.$$
The solid cha... | Hard | false | |
GT-0010 | Specify \(G\) as the connected orientable ribbon graph encoded by the conventions below. Treat the three panels of the supplied asset as the complete combinatorial specification of \(G\); no unshown vertex, crossing, edge band, or identification is present. In the left panel, every black point on a blue curve denotes a... | data/images/GT-0010.png | null | \(\left(x+1+xy+2y+y^2z^2,\;p^3t+3p^2qt^2+2pq^2t+pq^2t^3+q^3t^2\right)\) | 数学 | 图论 | Let \(u\) denote the upper-left black vertex and let \(v\) denote the lower black vertex. The left panel gives the basic sets and counts
$$V(G)=\{u,v\},\qquad E(G)=\{e_1,e_2,e_3\},\qquad |V(G)|=2,\qquad |E(G)|=3.$$
The two oval routes join the distinct vertices, while the handle-going route begins and ends at \(v\). ... | Medium | false | |
GT-0012 | Assume \(G\) to be the finite simple graph formed by the labeled corners and polygonal boundary sides of the given polyhedral presentation. Use the complete presentation as the incidence record for \(G\), including every visible portion of the solid. Each of the six parenthesized ordered pairs names the nearby geometri... | data/images/GT-0012.png | null | \((30,3,2)\) | 数学 | 图论 | The six corner labels supply the vertex set; the subscripts simply reproduce their ordered-pair names.
$$V(G)=\{v_{ij}:i\in\{0,1,2\},\ j\in\{0,1\}\},\qquad v_{ij}=(i,j).$$
Tracing the genuine terminal-to-terminal sides gives one triangular cycle in each second-coordinate layer and three sides joining corresponding co... | Medium | false | |
GT-0016 | We regard \(Q\) as the finite multigraph obtained from the given five-color presentation by the contraction and retention rule below. In that presentation, the color \(4\) edges are exactly the red strokes marked by repeated x symbols and labeled \(e_1,\ldots,e_7\), while the color \(3\) edges are exactly the black wav... | data/images/GT-0016.png | null | \((3,2,4)\) | 数学 | 图论 | Read the terminal junctions before contracting anything. Name the six junctions on the outer closed boundary \(a,b,c,d,e,f\) clockwise beginning at the far-left junction. On the left inner closed curve, assign \(g,h,i,j\) to the top-left, top-right, bottom-left, and bottom-right junctions, respectively. Use \(k,l,m,n\)... | Hard | false | |
GT-0018 | Here, define \(G\) as a finite undirected multigraph, and let \(L_G\) denote its integral combinatorial Laplacian. Construct \(G\) from the given directed presentation by the following closed-world extraction rule. Every named hand-drawn oval is one vertex; a small open or filled marker inside an oval is only decoratio... | data/images/GT-0018.png | null | \(\mathbb{Z}^{7}\oplus(\mathbb{Z}/4\mathbb{Z})^{6}\) | 数学 | 图论 | Write \(X-Y-\cdots\) for an undirected path, repeating the initial vertex when the listed edges form a cycle. Reading only the admissible oval-to-oval strokes gives seven connected components:
$$
G=\bigsqcup_{r=0}^{6}\Gamma_r.
$$
The upper-left component is a four-cycle with one path attached at \(I_{2,1,0}\):
$$
\G... | Hard | false | |
GT-0019 | The setup takes \(X=\{x_1,\ldots,x_{11}\}\) to be the ordered dart set encoded by the given strand presentation. Along the blue baseline, the dart labels occur in increasing subscript order from left to right. The marked point \(a\), located between \(x_6\) and \(x_7\), is only a separator and is excluded from \(X\). T... | data/images/GT-0019.png | null | \((5,\ t^4-9t-36)\) | 数学 | 图论 | The three upper green strands have rightward arrowheads. Following each smooth branch through its crossings gives
$$x_1\mapsto x_4,\qquad x_2\mapsto x_5,\qquad x_3\mapsto x_6.$$
The nested black strands above the baseline also point to the right. Their endpoint values are
$$x_4\mapsto x_{11},\qquad x_5\mapsto x_{10},\q... | Hard | true | |
GT-0049 | Introduce \(N\ge 10\) as an integer, and let \(X_N\) be the pulsed continuous-time simple exclusion chain defined below. The elevated finite network in the given presentation carries a standard graph-theoretic symbol. Interpret that symbol literally, and denote the resulting abstract graph by \(G_N\). The symbol fixes ... | data/images/GT-0049.png | null | $$\chi_{P_N}(x)=\prod_{j=0}^{5}\left(x-\exp\left(-\frac{j(N-j+1)}{\sqrt{N}}\right)\right)^{\binom{N}{j}-\binom{N}{j-1}}.$$ | 数学 | 图论 | The standard finite-network label identifies the complete graph:
$$
G_N=K_N,\qquad |V(G_N)|=N.
$$
Counting only the filled circular markers and reading the rate text give
$$
r=5,\qquad \rho_N=\sqrt{N}.
$$
The timing specification is
$$
I_k=\left[k\left(1+\frac{1}{N}\right),\;k\left(1+\frac{1}{N}\right)+\frac{1}{N}... | Hard | false | |
GT-0050 | Use \(G\) as the finite simple graph encoded by the colored straight-line network. Every filled circular marker is a vertex independently of whether its fill is green or yellow. Two distinct markers are adjacent exactly when a single drawn straight segment, green or yellow, can be traced with those two markers as its e... | data/images/GT-0050.png | null | 2^{20} * 3 * 5^4 * 7 * 11^5 * 17^3 * 467 * 593 * 661 | 数学 | 图论 | Independently traced black/colored marker endpoints give the 32-edge simple graph with edge list [['L1', 'L2'], ['L1', 'L3'], ['L1', 'L4'], ['L1', 'R1'], ['L1', 'Y1'], ['L1', 'Y2'], ['L2', 'L4'], ['L2', 'R2'], ['L2', 'Y1'], ['L2', 'Y3'], ['L3', 'L4'], ['L3', 'R3'], ['L3', 'Y2'], ['L3', 'Y4'], ['L4', 'R4'], ['L4', 'Y3']... | Hard | true | |
GT-0061 | In what follows, take $T$ to be the square cellulation of a torus with ten vertex rows and ten vertex columns, indexed by $\mathbb Z_{10}^2$. Regard the dashed square in the supplied display as one fundamental domain for $T$: opposite dashed sides are identified without a shift, so the left and right margin pieces of o... | data/images/GT-0061.png | null | (8,3,146) | 数学 | 图论 | The original image contains26 lavender plaquettes, including the entire periodic column5 and the visibly occupied cell(5,5) omitted by the old reference. The occupied plaquette coordinates are [[0, 2], [1, 5], [1, 7], [2, 1], [2, 2], [2, 3], [2, 7], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5], [5, 6], [5, 7], [5, 8]... | Hard | true | |
GT-0070 | Fix $G_1,G_2,G_3$ as connected undirected multigraphs defined by the curve-reading conventions below. Read the three separated drawings in the supplied asset from left to right. In each drawing, every solid black circular outline represents exactly one graph vertex. Its circumference and interior belong only to the ver... | data/images/GT-0070.png | null | $(2,2,48)$ | 数学 | 图论 | The positional convention fixes the vertex order in every drawing as
$$
(a_i,b_i,c_i,d_i)=(\text{upper-left},\text{upper-right},\text{lower-left},\text{lower-right}).
$$
In the left drawing, the long exterior strand begins at $a_1$ and terminates at $c_1$. It is therefore a second edge between those vertices. The compl... | Hard | false | |
GT-0081 | In this setup, consider \(G\) to be a finite connected undirected graph constructed by the centerline rules below. In the supplied planar trace, treat each of the three black vertical strokes and every blue stroke as a one-dimensional centerline arc. Ignore stroke thickness. The blank intervals in a dashed blue stroke ... | data/images/GT-0081.png | null | \((5,11,165)\) | 数学 | 图论 | Name the two transverse blue-blue crossings \(X\) and \(Y\), from left to right. Let \(P,Q,R\) be the three special branch meetings on the left, middle, and right black strokes, and let \(U,V,W\) be the ordinary blue-black crossings on those strokes. Removing all endpoint tails leaves a core \(\Gamma\) with vertex set
... | Hard | false | |
GT-0084 | For the discussion below, take \(D=(d_{ij})\) to be an \(8\times 8\) integer rank array whose entries lie in \(\{0,1,2,3\}\). Its row index increases from top to bottom, and its column index increases from left to right. Treat each open blue dashed square in the displayed grid as one array cell. Read the symbol centere... | data/images/GT-0084.png | null | \(\left(2^{23}\cdot 3^3\cdot 5^2\cdot 7,\;2^8\cdot 3^4\cdot 5^2,\;2^2\cdot 3^2\right)\) | 数学 | 图论 | Reading the cell symbols from top to bottom and left to right gives the rank array
$$D=\begin{pmatrix}1&1&1&1&1&1&1&1\\1&2&2&2&2&2&2&1\\1&2&3&3&3&3&2&1\\0&1&2&3&3&3&2&1\\0&0&1&2&3&3&2&1\\0&0&0&1&2&3&2&1\\0&0&0&0&1&2&2&1\\0&0&0&0&1&1&1&1\end{pmatrix}.$$
For \(q=1\), counting entries at least \(1\) in each row gives
$... | Hard | false | |
GT-0102 | The setup takes $T$ to be the finite reduced topological graph extracted from the given black-stroke presentation under the following closed rules. The straight horizontal and vertical coordinate axes, together with every tick and numerical coordinate label, are annotations and are deleted. In particular, the long near... | data/images/GT-0102.png | null | 3^14 | 数学 | 图论 | After deleting the coordinate axes and suppressing degree-two bends, the visible tree has30 vertices and29 edges, with14 trivalent internal vertices and16 leaves. In particular the upper-left junctions near(192,61) and(197,55), and lower-left near(192,521) and(197,527), are distinct pairs of Y-junctions connected by sh... | Hard | true | |
GT-0138 | Assume \(P\) to be the finite set of idealized centers of the large colored markers in the complete supplied square graphic, with the blue central aperture excluded. Count every gray disk, gold disk, and star-like glyph exactly once, and regard a small contrasting dot inside a larger glyph as decoration belonging to th... | data/images/GT-0138.png | null | (6,6) | 数学 | 图论 | The question explicitly asks to use the intended marker centers, ignoring glyph shapes and colors. All210 visible marker centers fit the triangular-lattice annulus {(i,j):9<=i^2+i*j+j^2<=64} around O. Root independently extracted210 centers and matched this integer point set one-to-one. The map R(i,j)=(-j,i+j) is a60-d... | Hard | false | |
GT-0163 | Use a finite planar cubical set \(K\) to determine a quotient space \(Y\) by collapsing all of \(K\) to one point inside the one-point compactification of the plane. Use the native pixel array of the supplied raster without smoothing, interpolation, curve fitting, or completion of interrupted strokes. For each pixel wi... | data/images/GT-0163.png | null | \(t+t^2\) | 数学 | 图论 | The color rule first isolates exactly the nonblack red support. In pixel notation, the active set is
$$
P=\{(i,j):r_{ij}>0,\ g_{ij}=0,\ b_{ij}=0\}.
$$
The associated planar cubical set is therefore
$$
K=\bigcup_{(i,j)\in P}[i,i+1]\times[j,j+1].
$$
Tracing adjacency through the visible red support shows one long compone... | Hard | false | |
GT-0164 | A connected graph \(H\) is constructed from a spoke-and-rim graph \(G\) and analyzed through its reduced Laplacian. On the supplied spherical rendering, locate the single small blue junction near the upper middle from which thin blue-gray smooth spokes travel toward the right-hand silhouette. Let \(N\) be the number of... | data/images/GT-0164.png | null | $(2,8,24)$ | 数学 | 图论 | The right-hand fan has three qualifying endpoints: one upper, one middle, and one lower. Thus the visual count is
$$N=3.$$
The spoke-and-rim construction consequently has the vertex and edge sets
$$V(G)=\{h,r_1,r_2,r_3\},$$
and
$$E(G)=\{hr_1,hr_2,hr_3,r_1r_2,r_2r_3,r_3r_1\}.$$
Every pair of these four vertices is... | Hard | true | |
GT-0167 | Throughout, use \(p\in S_{16}\) to denote the endpoint permutation of a system of sixteen connected strands, with both boundary rows indexed from left to right. Recover \(p\) from the complete supplied visual, which is indispensable data for this problem. Label the sixteen top terminals \(1,2,\ldots,16\) from left to r... | data/images/GT-0167.png | null | ((1,1,2,3,3,6),22,6) | 数学 | 图论 | The endpoint tracing in the complete visual gives the following values for the first four top terminals.
$$
p(1)=4
$$
The second strand terminates at the first bottom position.
$$
p(2)=1
$$
The third strand reaches the ninth bottom terminal.
$$
p(3)=9
$$
The fourth strand reaches the second bottom terminal.
$$
p(4)=2
$... | Hard | false | |
GT-0168 | We regard \(G\) as the finite undirected graph obtained from the given presentation by taking one vertex for each rounded blue box and one edge for each connector joining two such boxes. Use the complete displayed presentation as the data for this construction. The nine vertices are the boxes labeled [2]-manifolds, Sym... | data/images/GT-0168.png | null | 120 | 数学 | 图论 | Name the nine rounded boxes according to their visible labels as follows:
$$\begin{aligned}a&=\text{[2]-manifolds},&b&=\text{Symplectic [2]-manifolds},\\c&=\text{Lie 2-algebroids},&d&=\text{Poisson [2]-manifolds},\\e&=\text{Matched pairs of 2-representations},&f&=\text{Poisson Lie 2-algebroids},\\g&=\text{Degenerate C... | Medium | false | |
GT-0207 | Adopt \(\mathcal F\) as a finite family of plane trees whose vertices are represented by circular marks joined by straight strokes. The supplied asset specifies \(\mathcal F\) as follows: every maximal connected drawing separated from all other drawings by blank space is one member \(T\). Treat every circular mark, fil... | data/images/GT-0207.png | null | \((1557,\,2^{20}3^{15}5^3)\) | 数学 | 图论 | Every connected drawing in the asset is a tree on seven marked vertices. Consequently, each has six edges and total degree twelve:
$$
|V(T)|=7,\qquad |E(T)|=6,\qquad \sum_{v\in V(T)}d(v)=12.
$$
For a fixed vertex \(v\) of \(T\), all edges incident to \(v\) become mutually adjacent vertices in the line graph. They there... | Medium | false | |
GT-0208 | Start with \(C\) as the finite colored cycle formed by the complete outer circular chain of markers in the displayed tessellation. Traverse that entire outer chain clockwise, using every visible marker on the circumference, including solid black markers and thinly outlined white markers. Consecutive outer markers are a... | data/images/GT-0208.png | null | $$\left(\frac{x_1^{28}+3x_2^{14}+2x_1^2x_2^{13}+2x_4^7}{8},\frac{y_1^{56}+5y_2^{28}+2y_4^{14}}{8}\right)$$ | 数学 | 图论 | Read one complete quarter of the outer circumference clockwise, beginning at a cardinal white marker and ending immediately before the next cardinal white marker. The visible color word on this arc is
$$Q=W B^5 W B^2 W^4 B^2 W B^5.$$
The number of positions and the color counts in this word are
$$|Q|=21,$$
$$|Q|_W=7,$... | Hard | false | |
GT-0216 | The construction uses \(G\) as a finite planar polygonal cell complex whose bounded faces each contain exactly one marked site. In the complete visual asset, interpret every blue mark as one site and every closed gray polygonal region containing that mark as one bounded face. Interpret every visibly drawn gray segment ... | data/images/GT-0216.png | null | (102,271,578) | 数学 | 图论 | Two independent native-image counts identify102 blue sites, one in each of102 bounded Voronoi regions in the complete rectangle. Each side of the rectangle has nine boundary segments, so total boundary-segment count b=36. The four corner vertices have degree2 and every other graph vertex is trivalent under the given ge... | Hard | true | |
GT-0219 | The ordered Belyi passport encoded by the genus-one dessin in the supplied composite construction is the exact object to recover. Use the following conventions as a closed mathematical specification. Filled vertices represent points above \(0\), open vertices represent points above \(1\), and complementary regions repr... | data/images/GT-0219.png | null | ((6,3,3),(4,4,4),(2,2,2,2,2,2)) | 数学 | 图论 | Write the required ordered passport as
$$\mathcal P=(\lambda_0,\lambda_1,\lambda_\infty).$$
The solid and open vertex conventions determine which incidences belong to the first two partitions. In the local composite pattern, the solid quotient vertices are represented by A, F_1, and F_2. Their valencies read from the... | Medium | false | |
GT-0230 | Choose $G$ to be a crossing-incidence graph on six oriented strands. Read the displayed local tangle under the following closed conventions. A vertex of $G$ is one maximal oriented strand, with every short interruption at an ordinary crossing continued along the unique straight local trajectory. When tracing a strand, ... | data/images/GT-0230.png | null | $(2,8,24)$ | 数学 | 图论 | Tracing through the crossing gaps separates the six strands into three parallelism classes. Each class consists of two distinct antiparallel strands, so write
$$V(G)=A\sqcup B\sqcup C,\qquad |A|=|B|=|C|=2.$$
The displayed incidences show that strands in the same class never cross. Every strand in one class crosses each... | Hard | false | |
GT-0239 | Declare \(G\) to be the weighted multigraph defined by the six-panel arrangement and the rules below. The panels are read from left to right across the upper row, followed by left to right across the lower row. In panel \(i\), let \(r_i\) be the number of red arrowheads visibly present in that panel. Count arrowheads o... | data/images/GT-0239.png | null | 2808 | 数学 | 图论 | The six panels read row-major have red-arrowhead counts(1,1,2,5,3,4); the lower-left panel includes the small left-pointing arrow near native(66,286), so its count is5. Adding the stipulated single blue arrowhead gives bundle multiplicities(2,2,3,6,4,5) on the six-cycle. A spanning tree omits exactly one bundle and cho... | Hard | false | |
GT-0244 | We take \(G=(V,E)\) to be the finite labeled graph whose vertices are the circles in the given presentation, with each vertex carrying the five-character word printed beside it. Every vertex is visibly either filled or unfilled. Define \(D\) to be the set of edges represented by maximal blue dashed connections. A conne... | data/images/GT-0244.png | null | [8,"01325",31] | 数学 | 图论 | There are eight maximal blue dashed links. Their unfilled endpoint words are 01325,13025,01352,13052,13502,05132,50132,51302. Their inversion counts are1,3,2,4,5,4,5,7, summing31. The lexicographically least word is the five-symbol word 01325. Its leading zero is essential and must not be dropped or interpreted as inte... | Hard | true | |
GT-0248 | The setup takes \(H\) to be the vertex-disjoint union of six source graphs associated with Case 1, Subcase 2.1, Subcase 2.2, Case A, Case B, and Case C. The supplied composite drawing is part of the definition of these graphs and must be used to recover their adjacency data. In each named location, select the compact b... | data/images/GT-0248.png | null | P_H(q)=\left[q(q-1)^3\right]^3\left[(q-1)^4+(q-1)\right]^2\left[(q-1)^8+(q-1)\right] | 数学 | 图论 | The compact source graph in Case 1 is a central vertex joined to three leaves. The same visual pattern occurs in each of the two Case 2 subcases.
$$
H=G_1\sqcup G_{2.1}\sqcup G_{2.2}\sqcup G_A\sqcup G_B\sqcup G_C.
$$
Thus the first three components have the same graph isomorphism type.
$$
G_1\cong G_{2.1}\cong G_{2.... | Hard | true | |
GT-0255 | Introduce \(G\) as the simple graph with labeled vertices \(A, B, C, D, E,\) and \(F\), and let \(M\) be its graphic matroid. The supplied rendering is the complete combinatorial specification of \(G\): each black segment joining two labeled points is an edge, a dashed portion continues an occluded black segment, the t... | data/images/GT-0255.png | null | x^5+7x^4+28x^3+76x^2+139x+133 | 数学 | 图论 | Reading all solid and dashed black continuations in the rendering gives the twelve-edge set
$$
E(G)=\{AB,AC,AD,AE,BC,BD,BF,CE,CF,DE,DF,EF\}.
$$
The three pairs absent from this set are
$$
\{AF,BE,CD\}.
$$
Thus the vertices split into three nonadjacent pairs:
$$
\{A,F\},\qquad \{B,E\},\qquad \{C,D\}.
$$
Every vertex is ... | Hard | false | |
GT-0277 | Understand \(G\) to be the directed graph whose six vertices are the labeled boxes Identification, Adaptation, Estimation, Shared Controller, Human Operator, and System. The given presentation is the indispensable data for determining the directed arcs. Read every arrowhead and trace its connected wire from the boundar... | data/images/GT-0277.png | null | (7,10,t^6-3t^4-2t^3-t^2-2t-1) | 数学 | 图论 | The six box labels give the vertex set and the prescribed order immediately.
$$V(G)=\{I,A,E,C,H,S\}$$
The upper internal arrows give the three arcs linking Identification, Adaptation, Estimation, and Shared Controller.
$$\{(I,A),(E,I),(A,C)\}\subseteq E(G)$$
The branches leaving Shared Controller enter Estimation, ... | Hard | false | |
GT-0282 | A connected orientable ribbon graph \(J\) is specified by the middle band-and-disk presentation in the given arrangement. Use the left-hand panels labeled H1 and G2, the arrow labeled taking permissible join, the middle unlabeled presentation, and the right-hand panel labeled H2 exactly as arranged: these placements id... | data/images/GT-0282.png | null | (4,4,2) | 数学 | 图论 | The middle presentation has four gray disks. Reading the attachments around those disks gives a cyclic four-vertex ribbon graph. Name the disks in cyclic order v_1, v_2, v_3, v_4. Thus the vertex set is
$$V(J)=\{v_1,v_2,v_3,v_4\}$$
Therefore the first requested component is
$$v(J)=4$$
The four bands join consecutive ... | Hard | false | |
GT-0288 | Define the ordered three-row signature \(S=(s_1,s_2,s_3)\), where each row signature records three visual-combinatorial quantities extracted from a fixed three-column composite. Use the supplied complete composite as indispensable evidence, and read each horizontal band as one row with three linked panels. The top, mid... | data/images/GT-0288.png | null | [[2,1,2],[2,2,4],[2,7,3]] | 数学 | 图论 | Preserving top-to-bottom row alignment, each left panel has two continuous heavy black simple closed curves. The central panels contain respectively1,2,7 fully yellow triangular graph faces; in the sparse bottom graph all seven separately bounded yellow interiors count, while neighboring green triangles, unfilled cycle... | Medium | true | |
GT-0291 | Adopt \(G\) as the finite simple graph constructed from the thirteen bars in the supplied chart. The complete supplied bar chart is the sole numerical data source for this construction. Its vertical axis is uniformly scaled, with horizontal grid lines labeled 0, 20, 40, 60, and 80 percent. Index the bars from left to r... | data/images/GT-0291.png | null | (\lambda+1)^{571}(\lambda-76)(\lambda-70)^2(\lambda-57)(\lambda-53)(\lambda-47)(\lambda-43)(\lambda-39)(\lambda-37)(\lambda-35)(\lambda-32)(\lambda-7)(\lambda-5) | 数学 | 图论 | The grid spacing is 20 percentage points, so the vertical coordinate can be converted to a numerical height by linear interpolation. Reading the thirteen bar tops from left to right and applying the prescribed nearest-integer rule gives
$$0,20,40,60,80$$
The resulting ordered height sequence is
$$h=(77,71,71,58,54,4... | Hard | true | |
GT-0302 | Fix \(G_L\) and \(G_R\) as the plane graphs encoded by the left and right stroke networks, respectively, in the supplied two-panel drawing. The complete drawing is indispensable data: use every visible stroke, including the large outer returns, the elongated central loop, the lower return arcs, and every small lower en... | data/images/GT-0302.png | null | (0,5) | 数学 | 图论 | The explicit question defines the visible embedded stroke networks, includes every visible stroke, forbids outside features, and states that a white gap opening into the exterior is not a pocket. These rules do not authorize reconnecting invisible arcs across gaps. Ignoring knot over-under data likewise does not licens... | Hard | true | |
GT-0305 | View \(G\) as the undirected multigraph obtained from the six labeled panels and the direct arrow incidences in the supplied composite drawing. The vertices of \(G\) are the panels \(\mathrm{I}, \mathrm{II}, \mathrm{III}, \mathrm{IV}, \mathrm{V},\) and \(\mathrm{VI}\). Every visible red lowercase italic glyph is a cand... | data/images/GT-0305.png | null | X^5+X^4+X^3+X^2+X+Y | 数学 | 图论 | The question constructs an edge for a label appearing in exactly two panels when the panels also have a visible direct arrow between them. Independent full-image inventories give exactly-twice labels x,z,s,v,a,d, producing I-VI,I-II,II-III,III-IV,IV-V,V-VI. The labels y,r,b each appear in three panels and are excluded;... | Hard | false | |
GT-0317 | Write G for the unweighted simple graph with vertex set consisting of the fourteen buses labeled 1 through 14 in the complete network drawing. Every continuous blue transmission branch whose endpoints are bus terminals contributes one undirected edge between those two bus labels. A branch may bend or change direction a... | data/images/GT-0317.png | null | (3909,1,7) | 数学 | 图论 | Tracing all blue inter-bus endpoint branches gives the20 simple edges [[1, 2], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [4, 5], [4, 7], [4, 9], [5, 6], [6, 11], [6, 12], [6, 13], [7, 8], [7, 9], [9, 10], [9, 14], [10, 11], [12, 13], [13, 14]]. This edge list agrees with the list in the old explanation, but that explanat... | Hard | false | |
GT-0325 | Understand \(H\) to be the simple undirected adjacency graph of the four principal color domains represented by the complete colored cell array, and let \(\tau(H)\) be its spanning-tree count. The saturated domains are labeled \(R\), \(G\), \(B\), and \(Y\) according to their dominant red, green, blue, and yellow color... | data/images/GT-0325.png | null | 8 | 数学 | 图论 | The saturated color regions provide the four vertices, with the labels fixed by the statement.
$$V(H)=\{R,G,B,Y\}.$$
Global tracing of the red boundary shows direct positive-length contact with the green region and with the yellow region.
$$RG\in E(H),\qquad RY\in E(H).$$
The red and blue regions have no direct int... | Hard | false | |
GT-0361 | In this setup, consider \(G=(V,E)\) to be the finite embedded graph encoded by the complete colored drawing, with one vertex for every colored circular node and one edge for every visible straight segment joining two node centers. A crossing of two segments without a circular node does not create a vertex, and a line c... | data/images/GT-0361.png | null | (1,64,64) | 数学 | 图论 | The full drawing contains64 distinct bright-yellow circular nodes, visually organized in four height groups of16 each. Closely overlapping yellow circles have distinct boundaries and count separately; red, blue, orange, gray and purple circles are excluded. Height groups are not separate connected components: visible n... | Hard | true | |
GT-0372 | Introduce \(T\) as the oriented two-chain formed by the three green sheets in the upper pair of panels, and let \(B\) be the oriented chain represented by the two nearby green copies in the lower pair of panels. Each sheet carries the orientation indicated by its interior circular arrow, and each edge inherits the corr... | data/images/GT-0372.png | null | (4,2,1) | 数学 | 图论 | Corrected target: (4,2,1). Independent original-image and exact mathematical proof: {"root_full_four_panel_read":"In upper-right boundary graph the central singular point has four solid incident strata: toward upper-left neck, lower-left tip, upper-right junction, lower-right boundary. The lower-right construction show... | Hard | false | |
GT-0374 | Treat $D=[0,2]\times[3/200,3/50]$ as partitioned into five continuous phase strata denoted by $B,K,R,Y,$ and $W$. The supplied parameter plot is the sole evidence for determining the global incidence pattern of these strata. The symbols $B,K,R,$ and $Y$ correspond, respectively, to blue filled dots, black asterisks, re... | data/images/GT-0374.png | null | $\mathbb Z/3\mathbb Z\oplus\mathbb Z/3\mathbb Z$ | 数学 | 图论 | The vertices consist of the four marked phases and the connected white phase:
$$
V(G)=\{B,K,W,R,Y\}.
$$
The positive-length condition defining adjacency is
$$
UV\in E(G)\iff \mathcal H^1\!\left(\overline U\cap\overline V\cap\operatorname{int}D\right)>0,
$$
where $\mathcal H^1$ denotes one-dimensional length.
The lowe... | Hard | false | |
GT-0382 | Declare \(G\) to be the connected plane multigraph formed by the seven marked points and all curve segments in the supplied configuration. Use the complete drawing to label the five marked points on the large circular boundary, in clockwise order starting at the upper-left point, by \(a,b,c,d,e\). Label the central mar... | data/images/GT-0382.png | null | (10,11,620) | 数学 | 图论 | The original connected plane multigraph has seven vertices a,b,c,d,e,f,g and16 edges with multiplicity: ab,bc,cd,de,ea,ab,ae,af,af,ff,fg,fg,fb,fe,eb,ec. There are two distinct a-f arcs and a separate f-f loop; the gray disk boundary is excluded. The old reference omitted one a-f arc and the f-f loop. Hence cycle rank E... | Hard | true | |
GT-0385 | In this setup, consider \(Q\) to be the weighted directed quiver on the ordered vertex set \(V=\{1,2,3,4,5,6\}\), with the arrow directions and multiplicities encoded by the complete visual data supplied with this question. Every arrow without a printed multiplicity has multiplicity 1, while the visibly printed multipl... | data/images/GT-0385.png | null | [[-1,1,1,0,0,0],[1,1,0,1,0,0]] | 数学 | 图论 | The thirteen directed weighted edges are 1->3,1->4,2->1,2->3,4->2,3->4 (weight2),3->6,5->1,5->3,4->5,4->6,6->2,6->5; every unmarked edge has weight1. In particular the diagonal arrow points from3 to6, not6 to3 as the old reference records. With b_ij equal to arrows i->j minus arrows j->i, exact left-to-right RREF has r... | Hard | false | |
GT-0394 | Think of \(G\) as the planar graph formed by the union of the eight unit squares whose tile centers start at \((0,0)\) and make successive moves east, east, east, north, east, east, north. Every tile corner is a vertex, and every horizontal or vertical unit segment in the union is an edge. A side shared by two tiles is... | data/images/GT-0394.png | null | 3x^4y^5+13x^5y^4+16x^6y^3+4x^7y^2 | 数学 | 图论 | The square snake has column coordinates \(0,1,\ldots,6\). Reading the tile positions from the drawing gives
$$S=\{(0,0),(1,0),(2,0),(3,0),(3,1),(4,1),(5,1),(5,2)\}.$$
Only the tau or delta family is needed for the weight calculation, so the labels become
$$w(\tau_i)=x,\qquad w(\delta_i)=y.$$
The required enumerator i... | Hard | false | |
GT-0395 | Understand \(D\) to be the closed disk bounded by the large circle labeled \(R_1\), and let \(G\) be the embedded graph formed by the internal curves and all their marked endpoints. Every black dot and every arrowhead tip is a vertex of \(G\). A maximal drawn arc between consecutive marked vertices is one edge; ordinar... | data/images/GT-0395.png | null | (2,2,16) | 数学 | 图论 | Exclude the outer circle as prescribed. The connected core after suppressing degree-two subdivisions has13 vertices and14 edges, hence cycle rank2; the upper loop and left lobe are the two independent cycles. The complement inside the disk has three components, so reduced H0 rank2. Native-image arrow-tip census identif... | Medium | true |
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GraphTheory-VL
62 visual mathematics questions, including a nested 20-question hard subset. The questions connect diagrams to exact graph-theoretic, algebraic and topological targets. This repository contains original PNGs, exact question text, accepted reference targets and explanations, plus Hugging Face-compatible Parquet files with embedded image bytes.
Authors: Zixiong Yang, Kuo Zhou, Ruwei Pan, Jiaran Gao, Sihan Wu, Lu Zhang.
Affiliation: Key Laboratory of High Confidence Software Technologies, MoE, Peking University, Beijing, China.
Corresponding author: Lu Zhang, zhanglu@sei.pku.edu.cn.
Field definitions · Read without third-party packages · Data license · Citation
Project resources
Project homepage · Code and evaluation tools · Release v1.0
- Core dataset ZIP: portable questions, original images, prompts, recorded configurations, labels and offline evaluation tools.
- Complete response archive ZIP: 983 complete archived answers, effective judgments and scoring references for all 984 positions, including the one user-reported label without its complete answer.
The homepage includes real examples, recorded results and the draft paper. An arXiv identifier will be added when available.
Load the data
Use Python 3.10 or newer for the optional Hugging Face reader (verified with Python 3.12). The separate standard-library JSONL reader supports Python 3.9 or newer.
Install the optional dependencies:
python -m pip install -r requirements.txt
Load either evaluation configuration from Hugging Face:
from datasets import load_dataset
challenge = load_dataset("zixiong02/GraphTheory-VL", name="Challenge62", split="test")
hard = load_dataset("zixiong02/GraphTheory-VL", name="Hard20", split="test")
assert len(challenge) == 62
assert len(hard) == 20
example = challenge[0]
print(example["problem_id"], example["question"])
image = example["image"] # A decoded PIL image; no external image URL is needed.
From a downloaded or cloned copy of this repository, use its local directory instead:
from pathlib import Path
local_repository = str(Path(".").resolve())
challenge = load_dataset(local_repository, name="Challenge62", split="test")
hard = load_dataset(local_repository, name="Hard20", split="test")
Configurations and scope
| Configuration | Split | Rows | Relationship |
|---|---|---|---|
Challenge62 (default) |
test |
62 | All released evaluation questions |
Hard20 |
test |
20 | The exact 20 rows of Challenge62 with hard=true |
There are 62 unique questions, not 82 independent questions. The two test splits are alternative views of the same collection; they are not disjoint evaluation sets. No training or validation split is supplied. The original difficulty field is descriptive metadata; use hard or the Hard20 configuration for membership.
Challenge62 was selected from a 623-question source pool using model screening followed by reference examination and quality filtering. The remaining source-pool questions and mixed historical workbooks are outside this repository.
Files and image encoding
data/Challenge62/ and data/Hard20/ contain the Parquet test files. Their image column is a Hugging Face Image feature with the original PNG bytes embedded. Image decoding therefore works independently of the original storage service. Each row group contains one question, and the Parquet files include a page index. The images are preserved at original resolution, including the larger diagrams.
The portable source export is also retained: data/questions.jsonl, Challenge62 IDs, Hard20 IDs, and data/images/. These files are byte-identical to the verified v1.0 core package. The image_path field points to the corresponding local PNG in a full repository copy; Hub users can use the embedded image column directly.
Question, answer and explanation strings are unchanged. TeX notation remains text, options is null for all current questions, and categorical labels retain their original values. The Parquet rows add only the decoded-image feature to the portable JSONL fields. See SCHEMA.md.
Intended use and evaluation context
Use GraphTheory-VL to inspect diagram interpretation, exact mathematical answers, repeated success and the effect of removing visual inputs. Provide a solver with the question and intended image; reference answers and explanations are supplied for evaluation and inspection, not as solver input. Tasks include integer counts, tuples, abelian-group invariants and symbolic polynomials.
The recorded evaluation uses 744 original-image positions and 240 text-only positions across three models with four indexed responses per question. Its 984 unique positions include 983 complete archived outputs and one user-reported Incorrect decision without its complete local response. The responses and per-position evidence are available in the response archive; effective labels, prompts and evaluation code are in the code repository.
Reference checking combines recorded model-assisted inspection and exact computation; dataset-wide independent human validation has not been supplied. Selection conditions and the small Hard20 subset limit generalization. Target agreement alone does not establish a correct structural interpretation or derivation. GT-0560 preserves the wording used in the reported experiment, including its documented directional conflict.
Creation, licensing and attribution
The GraphTheory-VL research team created the 62 released question texts and accompanying images. These materials and team-owned dataset documentation use CC BY 4.0. Team-owned reading and verification code uses MIT. Model-service outputs and other third-party materials, where separately distributed, retain their applicable terms.
Use the six authors listed above and GraphTheory-VL, version 1.0 for attribution. The dataset citation in CITATION.bib links to the versioned release. The paper citation will be added when its identifier is available.
Local verification
With the optional dependencies installed, run:
python -m unittest discover -s tests -v
The checks load both configurations through load_dataset, confirm 62/20 memberships, compare every original field, verify the embedded PNG bytes, and decode all 82 configuration rows to compare pixels with the retained original images. They also check the standard-library reader from another working directory. The repeated 20 rows are verified as a nested subset.
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