Th powers and the problem of representing odd

th powers and the problem of representing odd integers as
sums of three primes. Hardy is also remembered for his collaborations with John E. Littlewood, a colleague at Cambridge, with
whom he wrote more than 100 papers, and the famous Indian mathematical prodigy Srinivasa Ramanujan. His collaboration with
LittlewoodledtothejokethattherewereonlythreeimportantEnglishmathematiciansatthattime,Hardy,Littlewood,andHardy–
Littlewood,althoughsomepeoplethoughtthatHardyhadinventedafictitiousperson,Littlewood,becauseLittlewoodwasseldom
seen outside Cambridge. Hardy had the wisdom of recognizing Ramanujan’s genius from unconventional but extremely creative
writings Ramanujan sent him, while other mathematicians failed to see the genius. Hardy brought Ramanujan to Cambridge and
collaborated on important joint papers, establishing new results on the number of partitions of an integer. Hardy was interested
in mathematics education, and his book A Course of Pure Mathematics had a profound effect on undergraduate instruction in
mathematics in the first half of the twentieth century. Hardy also wrote A Mathematician’s Apology , in which he gives his answer
to the question of whether it is worthwhile to devote one’s life to the study of mathematics. It presents Hardy’s view of what
mathematicsisandwhatamathematiciandoes.
Hardyhadastronginterestinsports.Hewasanavidcricketfanandfollowedscoresclosely.Onepeculiartraithehadwasthat
hedidnotlikehispicturetaken(onlyfivesnapshotsareknown)anddislikedmirrors,coveringthemwithtowelsimmediatelyupon
enteringahotelroom.

98 1/TheFoundations: LogicandProofs
Nonconstructiveexistenceproofsoftenarequitesubtle,asExample12illustrates.
EXAMPLE 12 Chomp isagameplayedbytwoplayers.Inthisgame,cookiesarelaidoutonarectangulargrid.
The cookie in the top left position is poisoned, as shown in Figure 1(a). The two players take
turnsmakingmoves;ateachmove,aplayerisrequiredtoeataremainingcookie,togetherwith
all cookies to the right and/or below it (see Figure 1(b), for example). The loser is the player
whohasnochoicebuttoeatthepoisonedcookie.Weaskwhetheroneofthetwoplayershasa
winningstrategy.Thatis,canoneoftheplayersalwaysmakemovesthatareguaranteedtolead
toawin?
Solution: We will give a nonconstructive existence proof of a winning strategy for the first
player.Thatis,wewillshowthatthefirstplayeralwayshasawinningstrategywithoutexplicitly
describingthemovesthisplayermustfollow.
First,notethatthegameendsandcannotfinishinadrawbecausewitheachmoveatleast
one cookie is eaten, so after no more than m × n moves the game ends, where the initial grid
is m × n . Now, suppose that the first player begins the game by eating just the cookie in the
bottom right corner.There are two possibilities, this is the first move of a winning strategy for
thefirstplayer,orthesecondplayercanmakeamovethatisthefirstmoveofawinningstrategy
for the second player. In this second case, instead of eating just the cookie in the bottom right
corner,thefirstplayercouldhavemadethesamemovethatthesecondplayermadeasthefirst