Walter Trump
Walter Trump (born 1953) is a German mathematician and retired high school teacher based in Nürnberg, specializing in recreational mathematics.[1][2] Trump's notable contributions include advancements in magic squares and cubes, geometric tilings, and square packings. In 2003, he collaborated with French software engineer Christian Boyer to discover the first perfect magic cube of order 5, the smallest possible such cube, where all rows, columns, pillars, space diagonals, and other broken diagonals sum to the same magic constant.[3][4] He has also explored ultra-magic squares, a class of pandiagonal magic squares with additional symmetry properties, and maintains extensive online resources documenting their construction and enumeration.[5] In the field of geometric puzzles and tilings, Trump developed a modified version of the Voderberg tile, a concave nonagon that allows two tiles to surround two others while all four touch, enabling monohedral tilings of the plane.[6] Additionally, in 1979, he improved upon earlier results by providing an efficient packing of 11 unit squares into a larger square of side length approximately 3.877, a configuration that has been independently verified and built upon in subsequent research.[7][8] His work, often shared through personal publications and collaborations, emphasizes computational and constructive approaches to these problems.[2]Biography
Early life
Walter Trump was born in Germany c. 1953. Little is known about his family background or specific childhood circumstances. At age 17 in late 1970, he began exploring recreational mathematics, creating puzzles in notebooks inspired by television programs featuring Prof. Heinz Haber's puzzles from the 1960s.[9] These early experiences ignited his passion for the field.Education
Walter Trump commenced his formal studies in mathematics and physics at the Friedrich-Alexander-Universität Erlangen-Nürnberg in 1973.[9][10] These dual disciplines provided the foundational training that aligned with his later career in education and his pursuits in recreational mathematics.[9] During his university tenure, Trump engaged with faculty on mathematical puzzles, including dissection problems he had begun exploring prior to enrollment, though professors cited time constraints for deeper investigation.[9] This interaction highlighted early academic influences that reinforced his inclination toward geometric and combinatorial challenges, building on pre-university inspirations from science programming.[9] No specific thesis or initial research projects from this period are documented in available sources.Professional career
Teaching roles
Walter Trump joined the Gymnasium in Stein, Bavaria, as a high school teacher of mathematics and physics in September 1985, continuing in this role until his retirement in August 2015.[10] His educational background in mathematics and physics prepared him for these positions, enabling him to deliver instruction at the upper secondary level in a German grammar school setting. Over the course of his three-decade career, Trump progressed to leadership roles, serving as Head of Department for mathematics and physics as well as Director of Studies.[1] These positions involved coordinating departmental activities, contributing to curriculum planning, and guiding student academic development within the school's framework.[1] A distinctive aspect of Trump's teaching approach was his incorporation of recreational mathematics to foster student interest and deeper understanding. He employed interactive puzzles and demonstrations, such as magic squares, to illustrate mathematical principles in an engaging manner, often using computer programs to explore concepts like number patterns that yield consistent sums.[11] This method aimed to captivate junior pupils and colleagues alike, transforming abstract topics into accessible and intriguing experiences without relying on traditional rote learning.[11] By blending such elements into regular lessons, Trump enhanced classroom dynamics and encouraged curiosity in the subjects he taught.[11]Retirement and legacy
Trump retired from his position as Director of Studies and Head of Department at Gymnasium Stein in 2015, enabling him to shift his focus entirely to independent research in recreational mathematics. Following retirement, he maintained active involvement in computational projects, often with assistance from his son Daniel, who contributed to calculations for perfect magic cubes, such as the computational search for order 5 in 2003.[12] Trump's enduring legacy centers on connecting formal education with accessible recreational mathematics, fostering interest among students, hobbyists, and researchers alike. His work has been recognized in the mathematical community for advancing fields like square packing and magic figures.[1] Additionally, his development of a hybrid backtracking Monte Carlo method has improved estimates for the enumeration of Franklin squares, influencing subsequent studies in magic square theory.[13] This post-retirement phase allowed Trump to extend the mathematical explorations that defined his teaching career, emphasizing innovative problem-solving accessible to diverse audiences.[1]Mathematical contributions
Square packing
Walter Trump's early contributions to recreational mathematics included significant work on the packing of equal squares into a larger square, particularly for the case of 11 unit squares. In 1979, he discovered the optimal known packing for this configuration, achieving a side length ratio of approximately 3.87708 for the enclosing square relative to the unit squares.[14][15] This arrangement features a near 2×3 grid where six squares are aligned orthogonally along the borders, while five inner squares are tilted at an angle of approximately 40.18° to fill a central gap efficiently, resulting in a rigid structure that prevents further adjustments without altering the overall geometry.[14][16] Trump's methodology relied on manual geometric analysis supplemented by calculations on a Hewlett-Packard 67 programmable calculator to solve the tilt angle equation, yielding a unique solution that optimized the packing density.[15] This approach marked a departure from prior restrictions to 0° or 45° tilts, demonstrating the value of arbitrary angles in enhancing efficiency for such problems. The discovery was later highlighted in Martin Gardner's writings, underscoring its elegance in recreational mathematics.[16] Compared to Friedrich Göbel's earlier packing from 1966, which used 45° rotations and required a larger side length, Trump's configuration represented a significant improvement, establishing a new benchmark for n=11.[14] This improvement highlighted the limitations of orthogonal and simple diagonal arrangements in square packing, influencing subsequent research into non-orthogonal tilts for higher n values. Computational searches as of 2025 confirm that Trump's packing remains unimprovable within its geometrical framework, solidifying its status as the densest known solution.[15]Magic squares
Walter Trump's research on magic squares encompasses advanced variants, including bimagic, trimagic, ultramagic, and subtraction types, where he employed computational methods to enumerate and analyze their structures. His work emphasizes enumeration, construction techniques, and unique properties such as constant sums under transformations like squaring or cubing. These contributions build on historical foundations while leveraging modern algorithms for exhaustive searches.[5] In 2014, Trump collaborated with Francis Gaspalou to enumerate all 8×8 bimagic squares, identifying a total of 26,158,848 such squares. Bimagic squares maintain magic properties—equal row, column, and diagonal sums—both in their original form and when each entry is squared, resulting in "double sums" that preserve the magic constant under this transformation. Their construction involved generating semi-bimagic precursors and applying targeted algorithms to filter full bimagic configurations, revealing patterns like those derived from Greco-Latin designs. This exhaustive computation, detailed in a joint paper, provided the first complete count and classified the squares into 192 fundamental sets, each expandable by symmetries.[17] Trump's investigations into trimagic squares, which remain magic under squaring and cubing, yielded significant discoveries for order 12. In June 2002, he constructed the first known 12×12 trimagic square, a rare achievement given the exponential growth in complexity for higher orders. These squares are exceptionally scarce; only a handful have been found, including subsequent examples derived by permutations and new constructions using specialized generation algorithms that explore additive and multiplicative symmetries. Among his contributions is a trimagic 12×12 square themed for the 60th anniversary of LEGO in 2017, incorporating celebratory elements while adhering to the triple magic constants of 870 (original), 29,490 (squared), and 1,404,300 (cubed). His methods relied on iterative computational searches starting from lower-order pandiagonals, highlighting the rarity as fewer than a dozen distinct variants are documented.[18] Trump also advanced the study of ultramagic 7×7 squares, which are both pandiagonal (magic in all broken diagonals) and associative (centrally symmetrical). Through detailed enumeration in 2004, he calculated a total of 20,190,684 such squares, each generating 49 panmagic variants via rotations and reflections. His analysis included algorithms for transposing and classifying these squares by cell coordinates and magic series, providing insights into their structural symmetries. Complementing this, Trump's work on subtraction squares—where differences between paired entries yield constant "residua"—rediscovered and translated historical examples, notably Adam Kochański's 1686 "quadrata subtractionis" of orders 3 and 4. For order 6, he enumerated 1,933 essentially different subtraction squares with residuum 15 that are also additive magic (sum 111), using filtered magic series from computational tools. These efforts underscore the interplay between additive and subtractive properties in early magic square designs.[19][20][21] Trump's website serves as a comprehensive resource, documenting enumeration counts for magic squares of various types across orders 6, 8, 10, and 12. For order 6, he reports 17,753,889,197,660,635,632 classic magic squares and 94,590,660,245,399,996,601,600 semi-magic ones, with zero associative, pandiagonal, or ultramagic variants. Order 8 features 5.2225 × 10⁵⁴ magic squares, 2.5228 × 10²⁷ associative, and 4.677 × 10¹⁵ ultramagic. For order 10, the totals are 2.4149 × 10¹¹⁰ magic and 1.4626 × 10¹¹⁵ semi-magic, again with no associative, pandiagonal, or ultramagic. Order 12 counts are not fully enumerated due to scale, but trimagic examples are highlighted as pioneering rarities. These figures, derived from his algorithms and collaborations, illustrate the vastness of the magic square landscape.[22]| Order | Magic Squares | Semi-Magic Squares | Associative | Pandiagonal | Ultramagic |
|---|---|---|---|---|---|
| 6 | 17,753,889,197,660,635,632 | 94,590,660,245,399,996,601,600 | 0 | 0 | 0 |
| 8 | 5.2225 × 10⁵⁴ | 1.0806 × 10⁵⁹ | 2.5228 × 10²⁷ | C8 + ? | 4.677 × 10¹⁵ |
| 10 | 2.4149 × 10¹¹⁰ | 1.4626 × 10¹¹⁵ | 0 | 0 | 0 |
| 12 | Not fully enumerated | Not fully enumerated | Not specified | Not specified | Not specified |