Is there a significant difference in calorie burn?
The brain, while only making up 2% of our body weight, accounts for ~20% of our energy use at rest. That's because the brain, being critical for survival, is a very high-maintenance organ. At rest, the membrane potentials of all neurons - firing or at "rest" - need to be controlled/maintained.
Of all the energy used by the brain, about 25% of it is needed for signal-independent function (protein synthesis, phospholipid turnover, etc.) Furthermore, there are two types of cells in the brain: neurons and glial cells. The glial cells (about 30%) use from 17 to 40% of the energy depending on whose estimates you use. The rest is needed (by neurons) for signal-related processes - maintaining/restoring cell membrane potential, etc. - only a fraction of which is used for thinking.
The brain never rests - it is highly active even in sleep, with both excitatory and inhibitory activities needing energy. Higher mental functions (e.g. doing math problems) only shifts the areas of highest activity a little bit compared to all the activity that occurs without our awareness of it.
Finally, if we could measure glucose consumption of the entire brain while daydreaming vs. doing math, we might be able to answer that question, but the truth is, brain activity doesn't occur in isolation. Someone doing challenging math problems is using more energy in other areas of the body as well; respiration rate, heart rate, muscle tension, etc. varies with the stress of the challenge, so glucose levels alone can't be relied upon (in fact, stress would increase our body's production of glucose) to indicate energy expenditure.
The end result is that in isolation, rigorous mental activity (which can be maintained only for a relatively short duration without fatigue, so it occurs in bursts) expends very little of the total energy the brain uses.
A common view equates concentrated mental effort with mental work, and it is fashionable to attribute a high demand for mental effort to the process of problem solving in mathematics. Nevertheless, there appears to be no increased energy utilization by the brain during such processes. From resting levels, total cerebral blood flow and oxygen consumption remain unchanged during the exertion of the mental effort required to solve complex arithmetical problems.
Cerebral Metabolic Rate in Various Physiological States
Does Thinking Really Hard Burn More Calories?