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Kate and Her Devoted Prince Are Bonding Over Baby

updated 04/10/2013 AT 5:00 PM ET

originally published 04/13/2013 AT 12:15 PM ET

From opposite ends of the Ping-Pong table, the famously competitive parents of Britain’s future monarch grinned at each other, and then let fly.

“Ahem, I think that was 6-2,” William said, declaring victory over Kate during their visit to a Glasgow sports center April 4. But, after his official duties were done, he turned to Lord Lieutenant Sadie Docherty, the couple’s host for the day, and chivalrously fessed up: “The only reason I won today was because Catherine was pregnant,” the prince told Docherty, who recounted the conversation to PEOPLE.

“Otherwise she would have taken off her coat and beat me.” That same day, while chatting with homeless youth at a Quarriers shelter, Will praised Kate’s cooking, pointing to his allegedly expanding waistline.

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