I urge you to do it using pencil and paper. You can solve this puzzle by performing a brute-force search with a computer, of course, but you don’t need to. For me as a puzzle maker, this digit substitution puzzle inspires the same feeling that Mozart inspired in Einstein, who said that Mozart’s music “was so pure that it seemed to have been ever-present in the universe, waiting to be discovered by the master.” Only someone as numerically gifted as Conway could have plucked such a perfect Platonic form from puzzle heaven! And then to have this natural set of conditions yield a unique answer is amazing. Once you read the first two conditions, you know exactly what the rest of puzzle is going to be. It flows completely naturally, without an iota of arbitrariness or artifice. Each of the digits is different, and they have the following properties:īefore you begin this puzzle, take a minute to admire the absolute perfection of its form.
There is a mysterious 10-digit decimal number, abcdefghij. Finally, we’ll immerse ourselves in an open-ended game contributed by a Quanta reader that resembles Conway’s iconic Game of Life. Then we’ll enjoy a geometric puzzle that relates to some of his most visually pleasing work. First, we’ll play around with a numerical puzzle Conway invented that is perfection itself. This month, we celebrate the playful genius of the famous British mathematician with two puzzles and an exploratory game. He made original contributions to group theory (the Leech lattice, monstrous moonshine), higher-dimensional geometry, tessellations, knot theory, number theory ( surreal numbers), algebra, mathematical logic and analysis. He invented the “ Doomsday algorithm” (a fast method of calculating the day of the week in your head - Conway could do it in under two seconds) and countless games, including Sprouts and the famous Game of Life, which launched the study of cellular automata.Ī great deal of Conway’s serious mathematical work also arose from his penchant for playing mathematical games. He performed detailed analyses of many puzzles, such as the Soma cube, peg solitaire and Conway’s soldiers. Such as b != 0 and g != 0, since numbers generally begin with a non-zero digit.The legendary mathematician John Horton Conway, who died in April of COVID-19, took a childlike delight in inventing puzzles and games. You can impose further constraints on the variables, if necessary. For example, y=0 was only explored as far as computing l=0 and immediately advancing to the next possible value for y. In actuality, fewer iterations will occur as computed values result in already used digits. Computing the unknown we find g=5, but that digit is already used, so we return to the previous level.ġ0 x 8 x 7 x 5 = 2800 loop iterations. The thousands would first be entered with y=1, l=0, o=2, r=3, b=5, i=4, a=9, carry=0. If a is a valid (unused) digit, compute the carry, and recurse into the thousands.
Then tens column is first entered with y=1, l=0, carry=0. If that doesn't find a solution, advance to the next value of y=2. At this point, we compute the carry (0), and recurse into the tens column. Starting with y=0, solving the equation gives l=0, which doesn't work, since 0 has already been assigned, so we immediately advance to the next value y=1, and again solve l=0. So you loop over the possibilities for y (0.9). In the above example, the ones column has y+l=y, so we start with 2 unknowns. Finally, for 3 unknowns, you loop over the two unknowns and compute the third. With 2 unknowns, you could loop through all possibilities for one of the unknowns, and compute the other. With one unknown, you could solve for the unknown directly. With 0 unknowns, you would only check if the equation was valid. Each set would compute the sum using module-10 arithmetic (or the base of your choice), and compute the carry into the next column.įor each set of equations, you could have between 0 and 3 unknowns. You would algorithmically solve this type of problem by dividing it into 4 sets of equations: the first for the ones column, the second for the tens column, the third for the hundreds column, the forth for the thousands column.
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