But even after being allowed to stand slightly closer to the target, their arrows went nowhere near, with Kate bursting into fits of giggles at her attempt and even appearing to apologise for her efforts.
With a large media pack looking vulnerable standing on one side of the 30 yard wide field, Kate too had a go with her bow.
'You'll be fine. You should try to shoot up in the air,' William told her. But her first effort only went about 20 yards, falling near but not dangerously near the royal press pack. 'Sorry, ' she said, giggling.
She and William were no more successful when they tried out Khuru, the nation's second most popular sport which was developed by cow herders who would fashion large darts out of tree branches and bird feathers.
They watched women competitors throwing the darts at a target, hoping to score a Karey - a hit.
But when Kate had a go she nearly scored a direct hit on a group of women players. Her dart fell just short of one of them and a second attempt went wide.
But her attempt was at least impressive due to the fact she was still wearing her beautiful handwoven Bhutanese-inspired outfit from her meeting with the King and Queen - and her cream Rupert Sanderson heels.
The royal couple, who passed up the chance to try out another Bhutanese sport - dae-gor, a version of boules played with rocks - were welcomed to the archery ground by the King's half-brother and national heartthrob, Prince Jigyel Ugyen Wangchuck.
After a cup of tea and snacks, he explained the rudiments of Bhutanese archery to the couple: the archers compete in teams based at either end of a 450ft long field, firing at wooden targets.
The men sing songs to celebrate hitting the target and are cheered on by women supporters who sing songs of support for their own men and obscene chants to try to distract their rivals.
William and Kate stood and watched as the archers showed off their skills, hitting the target from 450ft away, as a group of women sang soothing songs of support. 'That's amazing. That's the furthest I've ever seen anyone fire a bow and arrow,' he said.
When another archer stepped up, the women began singing short, louder verses, sounding more like football fans. 'Very good distraction songs,' William said, laughing.
But it was only after he had moved away that Sonam Karma, a Bhutanese Olympic Committee official who works closely with Pirince Jigyel explained just how rude some of the songs were about the competitors.
'They are singing obscenities,' he said, when a member of the press asked him to translate. 'You're bald. You're a baldy. Your nose is really big. We hope your private parts are as big.'
Many archers and their fans join in the traditional taunts but, as alcohol is also heavily consumed at archery competitions across Bhutan, it can lead to drunken brawls and archery hooliganism.
