By developing an approach by which we are able to quickly obtain spectra and eigenstates, we reveal for what is believed to be the first time the two novel phenomena of magic numbers and erratic level crossings in double-well Bose–Einstein condensates of N atoms. For $N⩽27$ and values of $U∕J$ that are not too small (U is the two-body interaction strength, and J is the hopping parameter), systems with even atoms are shown to be much more stable than those with odd atoms, and hence even integers play a role in such systems as if they were the magic numbers of nuclei. For $N⩾30$ , erratic level crossings occur as $NU⪢J$ .