Combinatorial Game Theory
who wins, and by how much
Generator

The simplest number in between

The simplest number in between
The simplest number in between. A game whose options are numbers is worth the simplest number strictly between them — and simplest means born earliest, so integers come before halves and halves before quarters. It is not the midpoint, and the difference is the whole content of the rule.

A game whose options are numbers is worth the simplest number strictly between them — and simplest means born earliest, so integers come before halves and halves before quarters. It is not the midpoint, and the difference is the whole content of the rule.

4 essays call simplicity. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

The positions it draws

31 distinct positions, harvested by drawing the figure again at the options each essay asked for.

PositionWorth OutcomeDrawn in
{−1 | 1} 0 P Cutcake, where every value is a whole number · Equal in every company
{−1/8 | 1/8} 0 P The simplicity rule
{−2 | 2} 0 P Maundy Cake · When a switch is not a switch
{−5 | 5} 0 P The simplicity rule
{−5 | 7} 0 P The simplicity rule
{−6 | 2} 0 P The simplicity rule
{0 | 1} 1/2 L A fight with no midpoint · Equal in every company · The simplicity rule · When a switch is not a switch
{0 | 1/2} 1/4 L A fight with no midpoint · The simplicity rule
{0 | 1/4} 1/8 P The simplicity rule
{0 | 1/8} 1/16 P The simplicity rule
{0 | 10} 1 P The simplicity rule
{0 | 2} 1 P Cutcake, where every value is a whole number · When a switch is not a switch
{0 | 3} 1 P The simplicity rule
{0 | 6} 1 P Maundy Cake
{1 | 3} 2 P Cutcake, where every value is a whole number
{1 | 4} 2 P The simplicity rule
{1 | 5} 2 P The simplicity rule
{1/4 | 1} 1/2 P The simplicity rule
{1/4 | 1/2} 3/8 P The simplicity rule
{1/4 | 3/4} 1/2 P The simplicity rule
{1/4 | 3/8} 5/16 P The simplicity rule
{2 | 11/4} 5/2 P When a switch is not a switch
{2 | 13/4} 3 L When a switch is not a switch
{2 | 4} 3 P Cutcake, where every value is a whole number
{2 | 5} 3 P When a switch is not a switch
{2 | 5/2} 9/4 L When a switch is not a switch
{3 | 20} 4 P Maundy Cake
{3 | 4} 7/2 P The simplicity rule
{3/2 | 7/4} 13/8 P The simplicity rule
{5/8 | 7/8} 3/4 P The simplicity rule
{7/8 | 9/8} 1 P The simplicity rule

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

The simplest number in between. A game whose options are numbers is worth the simplest number strictly between them — and simplest means born earliest, so integers come before halves and halves before quarters. It is not the midpoint, and the difference is the whole content of the rule. Values

The simplicity rule

When both players' options are numbers, the position is worth the simplest number strictly between them. Not the midpoint, not the average, and the difference between "simplest" and "middle" is the entire content of the rule.

 Cutcake: every value an integer. The value of an m by n cake, for every small m and n. Left cuts down, Right cuts across, and neither player ever gains by moving — so nothing is ever at stake, every value is a whole number, and the number says exactly how many spare moves one player has. Particular games

Cutcake, where every value is a whole number

A partizan game in which no position is ever worth a fraction, a star or a fight. Every value is an integer, the integer is a count of spare moves, and the pattern it follows is decided by binary digits.

 Maundy Cake: the pieces must be equal. The same cake as Cutcake, cut by the same two players, with one extra rule: a cut must divide the cake into equal pieces, and every piece stays in play. The values are still whole numbers, but the arithmetic that decides them is not Cutcake's — it counts prime factors rather than binary digits. Particular games

Maundy Cake

Cutcake with one word added: a cut must divide the piece into equal parts. The values are still whole numbers, and the rule this site has been repeating about them is false — over all 1,296 cakes to 36×36 the largest-odd-divisor account has 946 counterexamples. What survives is a count of prime factors, and it says who wins without saying by how much.

 Where a switch stops being a switch. The same Left option with the Right one raised past it. While the Left option is above the Right one both players want to move, the bar spans a real fight and the temperature is half the gap. Where the two meet the position is the number plus a star — no longer a switch, and not a number either. Above that the simplicity rule takes over: the value is the simplest number strictly between the options, and there is nothing to fight about. Values

When a switch is not a switch

A position {a | b} with numbers on both sides is a fight only while a is above b. Sweeping the boundary with a fixed at 2 and b climbing from −2 to 3 turns up three regimes rather than the two the definition suggests: eight fights whose temperature is exactly half the gap, one position at a = b that is 2∗ and is not a number, and beyond that numbers chosen by the simplicity rule — which on 8 of 12 sampled cases is not the midpoint.

The whole library · The position index · The figures that play back