Problem Statement

Jeff Tupper, in his 2001 paper "Reliable Two-Dimensional Graphing Methods for Mathematical Formulae with Two Free Variables", shows a set of methods to graph equations and inequalities with two variables in the cartesian plane. One of the examples of the paper, refers to the inequality:

1 2

<

⌊

m o d

(

⌊

y 17

⌋

2

− 17 ⌊ x ⌋ −

m o d

(

⌊ y ⌋ , 17

)

, 2

)

⌋

{\displaystyle {\frac {1}{2}}<\left\lfloor \mathrm {mod} \left(\left\lfloor {\frac {y}{17}}\right\rfloor 2^{-17\lfloor x\rfloor -\mathrm {mod} \left(\lfloor y\rfloor ,17\right)},2\right)\right\rfloor }

That inequality, once plotted in the range 0 ≤ x ≤ 106 and k ≤ y ≤ k + 17 for k = 960, 939, 379, 918, 958, 884, 971, 672, 962, 127, 852, 754, 715, 004, 339, 660, 129, 306, 651, 505, 519, 271, 702, 802, 395, 266, 424, 689, 642, 842, 174, 350, 718, 121, 267, 153, 782, 770, 623, 355, 993, 237, 280, 874, 144, 307, 891, 325, 963, 941, 337, 723, 487, 857, 735, 749, 823, 926, 629, 715, 517, 173, 716, 995, 165, 232, 890, 538, 221, 612, 403, 238, 855, 866, 184, 013, 235, 585, 136, 048, 828, 693, 337, 902, 491, 454, 229, 288, 667, 081, 096, 184, 496, 091, 705, 183, 454, 067, 827, 731, 551, 705, 405, 381, 627, 380, 967, 602, 565, 625, 016, 981, 482, 083, 418, 783, 163, 849, 115, 590, 225, 610, 003, 652, 351, 370, 343, 874, 461, 848, 378, 737, 238, 198, 224, 849, 863, 465, 033, 159, 410, 054, 974, 700, 593, 138, 339, 226, 497, 249, 461, 751, 545, 728, 366, 702, 369, 745, 461, 014, 655, 997, 933, 798, 537, 483, 143, 786, 841, 806, 593, 422, 227, 898, 388, 722, 980, 000, 748, 404, 719 produces a drawing that visually mimics the inequality itself, hence it is called self-referential. Although the inequality is intended to be drawn on the continuum of the cartesian plane, the drawing can be performed iterating over the integer values of both the horizontal and vertical ranges. Make a drawing of the Tupper's formula, either using text, a matrix or creating an image. This task requires arbitrary precision integer operations. If your language does not intrinsically support that, you can use a library. The value of k is an encoding of the bitmap of the image, therefore any 17-width bitmap can be produced, using its associated encoded value as k.

Let's start with the solution:

do -- plot Tupper's self-referential formula
   --[[    need to find x, y such that:
             1/2 < floor( mod( (y/17)*2^ - ( 17x - mod(y,17) ), 2 ) )
           where x in 0..106, y in k..k+16
   --]]
   local bint = require 'lua-bint-master\\bint'(2048)    -- need around 600 digits

   local k = bint.fromstring( "960939379918958884971672962127852754715004339660129306651505" ..
                              "519271702802395266424689642842174350718121267153782770623355" ..
                              "993237280874144307891325963941337723487857735749823926629715" ..
                              "517173716995165232890538221612403238855866184013235585136048" ..
                              "828693337902491454229288667081096184496091705183454067827731" ..
                              "551705405381627380967602565625016981482083418783163849115590" ..
                              "225610003652351370343874461848378737238198224849863465033159" ..
                              "410054974700593138339226497249461751545728366702369745461014" ..
                              "655997933798537483143786841806593422227898388722980000748404" ..
                              "719" )

    local b2     = bint.frominteger(  2 )
    local b17    = bint.frominteger( 17 )
    local kMod17 = bint.tointeger( k % b17 )
    for yDelta = 0, 16 do
        for x = 106, 0, -1 do
            local powerOf2  = - ( 17 * x ) - ( kMod17 + yDelta ) % 17
            -- if powerOf2 = 0, then ( v * 2 ^ powerOf2 ) mod 2 = v mod 2 
            -- if powerOf2 > 0, then ( v * 2 ^ powerOf2 ) mod 2 = 0
            if powerOf2 > 0 then
                io.write( " " )
            else
                local v = ( k + bint.frominteger( yDelta ) ) // b17
                if  powerOf2 < 0 then
                    v = v // bint.ipow( b2, bint.frominteger( - powerOf2 ) )
                end
                v = v % 2
                io.write( ( bint.eq( v, 0 ) and " " ) or "#" )
            end
        end
        io.write( "\n" )
    end
end