This paper assesses the feasibility of designing seed-and-blanket (S&B) sodium-cooled fast reactor (SFR) cores to generate a significant fraction of the core power from radial thorium-fueled blankets that operate in the breed-and-burn (B&B) mode. The radiation damage on the cladding material in both seed and blanket does not exceed the presently acceptable constraint of 200 displacements per atom (dpa). The S&B core is designed to have an elongated seed (or driver) to maximize the fraction of neutrons that radially leak into the subcritical B&B blanket and reduce the neutron loss via axial leakage. A specific objective of this study is to maximize the fraction of core power generated by the B&B blanket that is proportional to the neutron leakage rate from the seed to the blanket. Since the blanket feed fuel is very inexpensive and requires no reprocessing and remote fuel fabrication, a larger fraction of power from the blanket will result in a lower fuel cycle cost per unit of electricity generated by the SFR core. It is found possible to design the seed of the S&B core to have a lower transuranics (TRU) conversion ratio (CR) than a conventional advanced burner reactor (ABR) core without deteriorating core safety. This is due to the unique synergism between a low CR seed and the B&B thorium blanket. The benefits of the synergism are maximized when using an annular seed surrounded by inner and outer thorium blankets. Two high-performance S&B cores are designed to benefit from the annular seed concept: (1) an ultra-long-cycle core having a CR = 0.5 seed and a cycle length of ~7 effective full-power years (EFPYs) and (2) a high-transmutation core having a TRU CR of 0.0. The TRU transmutation rate of the latter core is comparable to that of the reference ABR with a CR of 0.5, and the thorium blanket can generate close to 60% of the core power. Because of the high blanket power fraction along with the high discharge burnup of the CR = 0 seed, the reprocessing capacity per unit of core power required by this S&B core is only approximately 1/6th that of the reference ABR core with a TRU CR of 0.5. Although the seed fuel CR is nearly zero, the burnup reactivity swing is low enough to enable a cycle length of more than 4 EFPYs. This is attributed to a combination of reactivity gain in the thorium blankets over the cycle and the relatively high heavy metal inventory. Moreover, despite the very low leakage, the S&B cores feature a less positive coolant reactivity coefficient and large enough negative Doppler coefficient even when using nonfertile fuel for the seed, because of the unique physics properties of the 233U and Th in the thorium blankets. With the long cycles, the S&B SFR is expected to have a higher capacity factor, and therefore a lower cost of electricity, than conventional ABRs. The discharge burnup of the thorium blanket fuel is typically 70 MWd/kg such that the thorium fuel utilization is approximately 12 times that of natural uranium in light water reactors. A sensitivity study is subsequently undertaken to quantify the trade-off between the core performances and several design variables: amount of zirconium in the inert matrix seed fuel, active core height, coolant pressure drop, and radiation damage constraint. The effect of the criterion used for quantifying acceptable radiation damage is evaluated as well. It is concluded that a viable S&B core can be designed without significant deviation from typical SFR core design practices.