Enigma 1126: Enigmatic dice
3 March 2017
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From New Scientist #2282, 17th March 2001 [link]
George has been winning free drinks at his local pub using a trick with four non-standard dice. Each face of each die is marked with one of the numbers 1 to 9, not necessarily all different. One of the nine numbers does not appear on any die, but each die has the same total of its six faces.
George allows you to choose one die, then he chooses one of the others. The two selected die are thrown simultaneously, and the one who throws the smaller number buys the drinks. Draws are impossible.
His friends have discovered that if they choose the red die, George chooses the yellow — if they choose yellow, George chooses green — if they choose green, George chooses blue — and if they choose blue, George chooses red! George expects (statistically) to win exactly two throws in every three with any of these pairs of dice.
We can conveniently represent the markings on a die as a six-digit number, with the digits in ascending order. You can check that 334455 beats 222288 two-to-one, but George’s set does not include either of these dice. The red die includes at least one lucky seven. There is only one set of four dice which will do the trick.
List the six numbers for each of the four colours.