divide

• person 1 values their own share at least as highly as person 1's ×9
• the value matrix, person 1's row is a list of 3 to 3 numbers ×7
• person 1's valuation is a list of 6 to 6 numbers ×6
• trimming piece 1 is worth exactly a third of the trimming to person 3 ×6
• the control matrix, person 1's row is a list of 3 to 3 numbers ×5
• each entry of person 1's valuation is a whole number between 0 and 100 ×4
• no allocation of these goods is envy-free — all 8 were tested and each failed ×4
• person 1 strictly prefers person 3's piece to their own ×4
• person 1's own piece is worth at least 25 to person 1 ×4
• person 1's valuation of the whole cake is 100 ×4
• person 1's values of the 4 pieces account for the whole cake ×4
• person 2's calling mark stays over the bar it belongs to ×4
• each entry of the control matrix, person 1's row is a whole number between 0 and 100 ×3
• each entry of the value matrix, person 1's row is a whole number between 0 and 100 ×3
• person 1's values of the three shares account for the whole cake ×3
• person 2 called the knife at exactly 25 ×3
• piece 1 is worth exactly a third of the cake to person 1 ×3
• the control matrix: person 1 values the whole set at 100 ×3
• the valuations are given for 2 to 2 people ×3
• the value matrix: person 1 values the whole set at 100 ×3
• a knife is asked to pass a non-negative amount of value ×1
• a rational is a whole numerator over a non-zero whole denominator ×1
• a rational is never divided by zero ×1
• every envy-free allocation is also envy-free up to one item ×1
• every person still waiting gets a calling mark ×1
• every way of giving each item to one of the people was formed ×1
• person 2 does not envy person 1's piece before the trimming is divided ×1
• person 4, who never called, is left with at least 25 ×1
• the chooser envies nobody, having taken the piece it valued higher ×1
• the chooser gains strictly more than half — the promise is not tight ×1
• the chooser's piece is worth at least half the cake to the chooser ×1
• the chooser's values of the two pieces sum to the whole cake ×1
• the claim that somebody is left envious is either made or not made ×1
• the claim that the chooser strictly gains is either made or not made ×1
• the control matrix gives a row of values for two, three or four people ×1
• the control was searched over every allocation too ×1
• the control: a chooser with the cutter's own measure gains exactly nothing ×1
• the control: the same search finds an envy-free allocation where one exists ×1
• the cutter's envy is exactly zero: the piece left behind is worth what the other is ×1
• the division being shown is one of cutchoose, trim, items, moving ×1
• the division is proportional and not envy-free: somebody prefers another's piece ×1
• the exact arithmetic stays inside the safe integer range ×1
• the exact value of the left piece and the decimal one agree ×1
• the number of items in the control matrix is a whole number between 2 and 6 ×1
• the number of items in the value matrix is a whole number between 2 and 6 ×1
• the number of segments the cake is cut into is a whole number between 2 and 12 ×1
• the round-robin allocation is envy-free up to one item ×1
• the three pieces go to three different people ×1
• the trimmed piece ties person 2's second largest exactly ×1
• the value asked for is still ahead of the knife ×1
• the value matrix gives a row of values for two, three or four people ×1
• two calling marks never print on top of one another ×1
• under the cutter's own measure the two pieces are exactly equal ×1

The whole library · What the figures prove