Mike Barker and John Rousseau met when they were both research physicists at Sheffield University, and kept in touch as each man moved into teaching the subject at Lancashire schools. Their introduction to Masquerade came on New Year's Day 1981, when their two families gathered at Rousseau's Fleetwood home. His three daughters, the oldest of whom was then 15, were working on the simple anagrams surrounding Masquerade's paintings, the adults joined in, and gradually John and Mike found themselves drawn into the book's deeper mysteries. “We'll be the ones to do this,” Rousseau told his old friend. “It needs a couple of physicists.”
Working together, the two men would eventually produce what Gascoigne calls “the perfect solution” to Masquerade. But they followed their own share of dead ends first. Rousseau, for example, stared at the Puppeteer picture long enough to conclude that the hat he wore was a beret - that is to say, he had a beret on his head. That sent John, his wife Sheila and his daughter Lizzie off on a 700-mile round trip to Berry Head in Devon, where the two grown-ups left Lizzie sleeping in the car while they went off into the night with their £50 metal detector.
“A policeman came along and he said ‘Hello, Love, where's your Mum and Dad?’” Rousseau recalled. “She said ‘They're out digging for treasure’. And he said ‘Oh, OK then’, and he wandered off. You can't believe it, can you? It was different world.”
All they unearthed that night was a stash of flattened baked bean tins, but even that expedition was not entirely wasted. It convinced Rousseau they must verify any theories carefully before embarking on another speculative trip. From now on, Barker and Rousseau would be strictly methodical in how they tackled the puzzle.
They had more luck with a special supplemental clue, published by The Sunday Times on December 21, 1980. This showed Williams' own self-portrait, surrounded by animals and clutching a fish in his right hand. Attached to the fish was a label reading “A6000”. Williams' left hand held up a piece of paper, covered in what looked to be nonsense symbols. Faithful to reality as always, he had drawn himself with the divergent squint which always made his eyes look as if they were pointing in two opposite directions.
The fish clue was quickly disposed of by a couple of physicists: 6,000 Angstroms (commonly abbreviated to “A6000”) represents the red wavelength of the spectrum. The fish itself was clearly a herring, and red herrings could be discarded at once. After a good deal of folding and reflecting in mirrors, the half-characters Williams had used on his paper message combined to read: “2 do my work IA.ed IV men from XX. The tallest and the fattest and the righteous follow the sinister”.
Not realising that Williams' squint was a simple representation of how he happened to look, Rousseau concluded they should think in terms of lines extended outward from his eyes to the edges of the picture. As with Parrack's theories about gems in eyes and the supposed role of Needlehole Farm, this was another case of someone stumbling close to the right answer for entirely the wrong reasons.
Putting the squint theory to one side for a moment, Barker and Rousseau tidied up Williams' code phrase to read: “To do my work, I appointed four men from twenty: the tallest and the fattest. And the righteous follow the sinister.” Four men from twenty suggested four fingers or toes from the twenty which each individual has. The tallest and fattest of these would be the two middle fingers and the two big toes. If the righteous followed the sinister, then that must mean the left hand (or foot) came before the right. This didn't seem to mean much on its own, but the two men noted down their findings and waited to see how they might fit into the bigger picture.
Their next step came when they spotted the similarity between the two magic squares Williams had included in the book - one in his Pennypockets painting, the other in his Puppeteer one. A magic square is a grid of numbers which produce the same total when added vertically, horizontally or diagonally. In a 4x4 square like the ones Williams used, each corner block of four numbers should produce that same total too. Williams followed these rules exactly with the Pennypockets square, producing a consistent total of 34 everywhere you looked, but added a twist of his own by leaving one entry blank. The Puppeteer's square used letters instead of numbers, and printed those letters in several different colours. Again, it left a single entry blank.
Barker and Rousseau had already been fooled by the atomic numbers in Williams' playing field square - “False: Now think again” - so this time they looked at how the two remaining squares might relate to one another. The two blank entries were each in the same position - third row down, third column along - so it did look as if the two squares were meant to be read together.
Matching the numbers from the Pennypockets square to the colours used in their matching Puppeteer entries produced an intriguing result. In the Pennypockets square, for example, the top left corner was occupied by a number 16, and the square next door by a 3. The equivalent two positions on the Puppeteer square are occupied by letters in blue and green respectively. Mapping each number against its matching colour and then re-arranging the numbers in ascending order produced a sequence of “red, yellow, green, blue” which repeated itself until the square was complete.
Pressing on, the duo then turned to the coloured rings on the puppeteer's fingers, each one connected to a string operating his two puppets. Applying their colour sequence, they noted that each of the red rings was connected to the middle finger on a puppet's left hand, each of the yellow ones to a left big toe. The green rings always linked to a right middle finger and the blue ones to a right big toe. That gave a new sequence, this one reading “left middle finger, left big toe, right middle finger, right big toe”. That fitted the Sunday Times clue precisely, which meant it must surely be right.
The girl puppet was holding both hands to her eyes, and that served as a reminder of Williams' introductory lines to the book: “To solve the hidden riddle, you must use your eyes”. The squint in Williams' self-portrait had already started Rousseau thinking about lines extending from a character's eyes to the painting's lettered frame, and the fact that those letters seemed so unevenly spaced suggested this theory was sound.
The two men had already decided that only a piece of written information could identify the hare's location with enough precision. Pointing his readers towards certain letters in the pictures' frames would certainly be one way for Williams to do it. But which lines from which character's eyes would he have used? And drawn through which other points on the painting to reach the particular letters required? Again, it would have to very precise to guarantee the right result.
The breakthrough came when Rousseau sat down with a ruler to study the picture showing a ring of animals circling the sun. Each animal in this picture displays just one eye and a single foot - always the right foot of its front legs. This simplified matters considerably, and Rousseau started drawing lines from each animal's eye through the furthest point on its front hoof or paw. This, he reasoned, was the closest equivalent any animal could have to a middle finger. He began with the hare, worked his way clockwise round the circle of six animals, and soon started getting results.