Joe Duffy is going round in circles. He seems to do little else these days

Usually, the information that Joe Duffy recently spent an hour going around in circles would hardly be noteworthy; after all, there are days when he seems to do little else on Liveline (RTÉ Radio 1, weekdays). But it turns out that Duffy has also been making circular motions in a more literal sense, as he informs listeners on Wednesday. “For some reason I bought a geometry set last week, because I wanted to do what I failed in school,” he says. “I was out with the compass the other night and I got an hour of concentric circles.”

It’s a moot point whether Duffy needs to revisit his geometry – as a radio host, he’s always able to find an angle – but as last week underscores, there are people who still find use for their old maths lessons. After a caller’s casual remark about never using the algebra in everyday life, Lee is moved to expound on its practical applications. He explains that he uses algebra to bet on the Lotto in bookies, with apparent success: he claims to have won €50,000 this year alone.

His host is incredulous that the supposedly random lottery numbers should come up in the regular patterns that Lee purports to find. Moreover, Duffy seems faintly nervous about encouraging such flutters. “You know there are TDs who want to ban betting on the Lotto in bookies,” he says, by way of feeble pushback.

He sounds happier the next day, when Mick holds forth on the joys of trigonometry, from its application in marking out pitches to the pure mathematical wonder of pi. Duffy responds enthusiastically, revealing his geometry set purchase and generally hamming it up. “You love pi,” he yuks, with the glee of schoolkid discovering the number’s baked good-related pun potential for the first time.

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Duffy is palpably moved, allowing himself a rare moment of patriotic optimism: ‘I think that at the heart of it, we are a good and decent country’

Duffy’s good mood isn’t only down to his newly rediscovered interest in hard sums. He has earlier rejoiced at the news that the HSE has approved the drug Kaftrio for children with cystic fibrosis. One caller, also named Joe, chokes up at the prospect of the drug transforming his granddaughter’s life. “It’s like winning the Lottery twice,” he says. (Lee might have some tips on how to do that.)

Duffy also speaks to seven year-old Ciaran, who has the condition. Ciaran says now that the life-changing drug is available, he can finally work toward his dream of playing for Manchester United; his delight is doubled when he’s offered tickets to Old Trafford by another caller, Paul. It’s such a perfectly aligned moment that one might dub the show Joe’ll Fix It, if it didn’t have such deeply creepy associations. Either way, Duffy is palpably moved, allowing himself a rare moment of patriotic optimism: “I think that at the heart of it, we are a good and decent country.”

[ CF drug approved after lengthy delay caused by price dispute ]

[ Joe Duffy brings us the upmarket tabloid of Irish radio ]

In stark contrast, Duffy also talks to Olga, who tells of life in the Russian-occupied Ukrainian city of Mariupol. She is unsurprisingly scathing about last week’s visit to the ravaged city by Vladimir Putin, or as Duffy contemptuously calls him, “the war criminal Pyoootin”. A journalist by trade, Olga – not her real name, the host stresses “for the benefit of Russian spies” in the Dublin embassy – is sober and honest in her account.

Though much of Mariupol was destroyed was Russian forces, she says her own home was untouched, and the supermarkets are stocked with food. But people are afraid: she compares life in the city to having “a gun beside your head”. “To be honest, I risk very much, even when I talk to you,” she says, in a rare flash of anxiety. Clearly pained by what he’s hearing, the host sounds atypically hesitant, but it’s an unvarnished reminder that Russia’s aggression in Ukraine continues unabated. Whatever about his geometric doodling, Duffy’s moral compass is as sharp as ever.

Over on Lunchtime Live (Newstalk, weekdays), host Andrea Gilligan puts a different spin on her afternoon phone-in show, preferring a spirit of inquiry to Duffy’s self-assured stances. “I want to know how bad this is,” she asks, opening Tuesday’s conversation on racism. Pretty awful, answers Nigerian-born Limerick resident Toby, who tells of ignorant teachers and physical intimidation while attending school in the 1990s. Gilligan sounds genuinely taken aback by such experiences, wondering if things have since improved. “It’s very hard to quantify,” Toby says, “But from my perspective, I know when I wake up every day that I have to face prejudice, unconscious biases and stereotypes.”

If Gilligan’s ingenuous manner sometimes comes across as wide-eyed, she also possesses an understated, practical side, which in this case helps to steer the otherwise glum item gently in a constructive direction

Gilligan listens intently, but is also keen to find solutions. “I’m trying to get an insight into how do you bring about change,” she asks of college student Eric, a co-ordinator for the Black and Irish organisation. Eric has his own tales of prejudice, but also sees “intercultural spaces” and increased political representation as paths to a new – and better, more decent – Ireland.

If Gilligan’s ingenuous manner sometimes comes across as wide-eyed, she also possesses an understated, practical side, which in this case helps to steer the otherwise glum item gently in a constructive direction. Whether the positive outcome she seeks will occur is another matter, but the item underscores her unfussy empathy.

It’s a quality again on show on Wednesday as she chats with celebrity chef Joe Shannon, who tells the host of his diagnosis with terminal cancer. Despite this dreadful news, Shannon is determined not to be downcast. “I want to enjoy the rest of my life,” he says. “I want positivity around me, the last thing I want is pity.” Shannon’s outlook is humbling but also inspiring, making for an oddly upbeat item, as the host praises her guest with typically low-key generosity: “You’ve a really pragmatic approach.” Her amiable curiosity notwithstanding, Gilligan likes to get to the point.