A modular block construction system comprises blocks and other accessory parts that dynamically fit together to form different items of varying forms and shapes. One or more blocks of various sizes have outer, peripheral edges that can be mated with one another, or snap fitted to accessories including walls and wedges. Each cubicle block comprises six intersecting panels that have peripheral edges that alternate with pairs of spaced apart nubs and sockets that enable the panels to be orthogonally assembled. Outwardly projecting panel sides of assembled blocks can be snap fitted to abutting sides of other blocks. Spider connectors captivated within the blocks transform each connecting surface between male, female, and flattened configurations. Radially spaced apart follower slots are defined in the sides. Movable spiders captivated within the blocks have arms slidably received by and constrained within the slots. The spiders are exteriorly accessible by the user who can touch exposed spider arm tips projecting from the block sides through the slots. Each panel comprises a plurality of identical quadrants separated from one another by the elongated follower slots. The spiders each have projections that variably penetrate orifices in the block panels.