Two games that are secretly one
Tonight's idea
JAM is a game for two, described by the Dutch psychologist John Michon in 1967. The map below has eight towns and nine straight roads. Players take turns to claim a road. The first to own three roads that pass through the same town wins.
It looks like a new game. It is not. Write the nine road numbers in a 3 by 3 grid, as on the right of the figure. Now the three roads through any town sit in a line: a row, a column or a diagonal. So claiming roads in JAM is marking cells in tic-tac-toe (noughts and crosses).
An isomorphism is a matching between two systems that pairs every piece of one with exactly one piece of the other and keeps the structure. Here each road is paired with a cell, and three roads share a town exactly when their three cells make a line.
Isomorphic games are one game in two costumes. Whatever is true of one is true of the other, once you translate.
Watch one: translate a game
Towns are lines. Town D is on roads 2, 9 and 4. In the grid those are the left column. Roads are cells. Road 5 passes through four towns, B, H, E and G. Cell 5 is the centre, which lies on four lines.
A game. Ann claims road 5, Ben 4, Ann 6, Ben 9. In the grid Ann has the centre and the top right corner. Ben has two cells of the left column, so he threatens the third, cell 2. On the map: Ben owns two of the roads through town D and needs only road 2.
Your turn
- (a) For each of the eight towns, list its three roads from the map and say which line of the grid they make. (b) How many towns does each road pass through? Which cells match the roads with two towns, with three, and with four?
- Carry on the game from the example. Ann owns roads 5 and 6, Ben owns 4 and 9, and it is Ann's turn. (a) Which road must Ann claim now, and why? (b) After that move, Ann threatens two towns at once. Which two, and why can Ben not stop her?
- Tic-tac-toe has been checked completely: if neither player makes a mistake, the game is a draw. (a) What does that tell you about JAM? Explain how you know without playing a single game of JAM. (b) Many tic-tac-toe players open in the centre. Which road is that, and what is special about it on the map?
- A game called Q has nine cards and eight winning sets of three cards. No card is in more than three winning sets. Can Q be tic-tac-toe in a third costume? Say what an isomorphism would have to do, and why it cannot.
Stretch: a third costume
Nine cards numbered 1 to 9 lie face up. Two players take turns to pick one. The first player holding three cards that add to exactly 15 wins. (a) Add the three road numbers at each town of the map. (b) Find every set of three different numbers from 1 to 9 that adds to 15. (c) Explain why this card game is isomorphic to JAM, and so to tic-tac-toe.
Answers
- (a) A: 2, 7, 6, the top row. B: 9, 5, 1, the middle row. C: 4, 3, 8, the bottom row. D: 2, 9, 4, the left column. E: 7, 5, 3, the middle column. F: 6, 1, 8, the right column. G: 2, 5, 8, a diagonal. H: 6, 5, 4, the other diagonal. Eight towns, eight lines. (b) Roads 1, 3, 7 and 9 pass through two towns and match the middle cells of the four sides. Roads 2, 4, 6 and 8 pass through three towns and match the corners. Road 5 passes through four and matches the centre.
- (a) Road 2. Ben owns 4 and 9, and roads 2, 9, 4 all pass through town D, so Ben would win with road 2. Ann cannot win at once, because the only town with both 5 and 6 is H, and its third road, 4, is Ben's. (b) Ann now owns 2, 5 and 6. Town A is on roads 2, 7, 6, so she needs 7. Town G is on roads 2, 5, 8, so she needs 8. Roads 7 and 8 are both free, Ben can claim only one, and no single road wins the game for Ben first. Ann takes the other and wins. In the grid she has made a fork: the top row and a diagonal.
- (a) JAM is also a draw when neither player makes a mistake. Any game of JAM can be translated, move by move, into a game of tic-tac-toe with the same winner, and back again. So if one player could force a win in JAM, the translated moves would force a win in tic-tac-toe, and that is impossible. (b) Road 5. It is the only road through four towns, so it can help to complete more towns than any other road.
- No. An isomorphism would pair each card with one cell so that winning sets match lines exactly. Then every card would be in as many winning sets as its cell is in lines. The centre cell is in four lines, so some card would have to be in four winning sets, and Q has none. Counting like this is a quick way to show that two systems are not isomorphic.
Stretch. (a) Every town gives 15: for example \(2 + 7 + 6\), \(2 + 9 + 4\) and \(6 + 5 + 4\). (b) There are exactly eight: 1, 5, 9; 1, 6, 8; 2, 4, 9; 2, 5, 8; 2, 6, 7; 3, 4, 8; 3, 5, 7; 4, 5, 6. To be sure none is missing, take each smallest number in turn: with 1 the other two add to 14, with 2 to 13, with 3 to 12, with 4 to 11, and a smallest number of 5 or more gives at least \(5 + 6 + 7 = 18\). (c) The eight sets in (b) are exactly the eight towns' sets of roads from question 1. So pair card \(n\) with road \(n\): three cards add to 15 exactly when their three roads share a town. The card game is JAM, and JAM is tic-tac-toe. The grid in the figure is a magic square, which is why every line adds to 15.