My buddy came up with a good question that I'm not sure I know the answer to.
Much like how back in the day when you announced that you were casting Thoughtcast with a few artifacts on the battlefield, one of them being a Chromatic Sphere, you reduce the costs appropriately to find out how much you pay to put the spell on the stack, then when you go to pay those costs, you could crack the Star to produce the blue mana you needed to help cast the Thoughtcast.
I'm not sure the same way applies here, but if my buddy has a food artifact and a Gilded Goose on the battlefield and he wants to cast Whir of Invention, can he tap the food to have improvise help pay for part of the cost of Whir of Invention and still be able to sacrifice the food to Gilded Goose to also help pay for the Whir's cost? I'm thinking yes, but I want to make sure before I tell him he can or can't.
I'm guessing a judge of Level 2 or higher might need to answer this with certainty.
In general, as you cast Whir of Invention, you have the chance to activate mana abilities before you pay its costs (C.R. 601.2g-h). If you activate Gilded Goose's last ability for mana as you cast that spell, the Food you sacrifice this way won't be there to be tapped by the time you pay Whir of Invention's costs (C.R. 601.2g-h, 702.125a, 205.3g, 605.1a, 701.16a). (Note that if X is 0, Whir of Invention's total cost will be UUU and so have no generic mana in it, so no artifacts can be tapped to pay for that spell [C.R. 601.2f, C.R. 601.2h, 702.125a, 107.4b]. Remember that to pay for a spell with improvise, you can tap no more artifacts you control than the amount of "generic mana in [the] spell's total cost" [C.R. 702.125a; under that rule, you tap the artifacts "rather than pay" the corresponding generic mana].)
Note that improvise isn't a cost reduction effect (unlike affinity for artifacts), but rather it changes the way you pay for the spell with that ability (compare C.R. 702.125a with C.R. 702.40a; see also C.R. 702.125b).
can he tap the food to have improvise help pay for part of the cost of Whir of Invention and still be able to sacrifice the food to Gilded Goose to also help pay for the Whir's cost?
No, he can't.
1) Mana must be generated prior to actual payment. (rule 601.2g)
2) Generating mana with Goose makes Food be sacrificed.
3) When time comes to actually pay (rule 601.2h), Food is gone; cannot be tapped to improvise.
601.2f The player determines the total cost of the spell. (…)
601.2g (…) Mana abilities must be activated before costs are paid.
601.2h The player pays the total cost (…)
Much like how back in the day when you announced that you were casting Thoughtcast with a few artifacts on the battlefield, one of them being a Chromatic Sphere, you reduce the costs appropriately to find out how much you pay to put the spell on the stack, then when you go to pay those costs, you could crack the Star to produce the blue mana you needed to help cast the Thoughtcast.
I'm not sure the same way applies here, but if my buddy has a food artifact and a Gilded Goose on the battlefield and he wants to cast Whir of Invention, can he tap the food to have improvise help pay for part of the cost of Whir of Invention and still be able to sacrifice the food to Gilded Goose to also help pay for the Whir's cost? I'm thinking yes, but I want to make sure before I tell him he can or can't.
I'm guessing a judge of Level 2 or higher might need to answer this with certainty.
Note that improvise isn't a cost reduction effect (unlike affinity for artifacts), but rather it changes the way you pay for the spell with that ability (compare C.R. 702.125a with C.R. 702.40a; see also C.R. 702.125b).
See also this thread, this thread, and this thread.
1) Mana must be generated prior to actual payment. (rule 601.2g)
2) Generating mana with Goose makes Food be sacrificed.
3) When time comes to actually pay (rule 601.2h), Food is gone; cannot be tapped to improvise.
601.2g (…) Mana abilities must be activated before costs are paid.
601.2h The player pays the total cost (…)
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules