Reverse engineering cash: Coin designs mark out high value differentials and coin sizes track values logarithmically.

Coins are physical representations of monetary values. Like mental or verbal representations of quantities, coins encode sums of money in formats shaped, in part, by cognitive and communicative needs. Studying the coins circulating today, we consider how their design, colour, and size reflect their value. We show that coin designs solve a trade-off between informativeness-the pressure to highlight distinct denominations-and simplicity-the pressure to limit the number of designs that coin users must memorise. Coinage worldwide is more likely to display distinctive graphic designs and distinct colours on pairs of coins with large differences in value, thus minimising the aggregate cost of mistaking one coin for another. Coin size differentials, in contrast, do not seem to indicate greater value differentials, although absolute coin sizes do reflect monetary values. Log-transformed values predict design and colour distinctiveness in coin pairs, as well as absolute coin sizes, better than raw values, consistent with research suggesting that monetary quantities may recruit the "numerosity system" for magnitude representations, thought to track quantities logarithmically. These results show that coins obey similar informational constraints as linguistic and mental representations.